'€ 


V 


h 


II 


Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/elementsofplanesOOcrocrich 


o 


O^qIu 


0  AX 


ELEMENTS 


OF 


PLANE   AND   SPHERICAL 


TRIGONOMETRY 


BY 


C.    W.    CROCKETT 

PROFESSOR  OF  MATHEMATICS  AND  ASTRONOMY 
RENSSELAER   POLYTECHNIC   INSTITUTE 


>>Hc 


NEW  YORK  :•  CINCINNATI  : •  CHICAGO 

AMERICAN    BOOK    COMPANY 


y\  /<;- 


Copyright,  1896,  by 
AMERICAN  BOOK  COMPANY. 

OBOCKXTT.       PLANE  AND  SPHES.   TBIGONOM. 
W.    P.    2 


PREFACE. 


This  work  has  been  prepared  for  the  use  of  beginners  in 
the  study  of  trigonometry.  Assuming  that  a  high  degree  of 
proficiency  cannot  be  expected  from  such  students,  the  author 
has  limited  himself  to  the  selection  of  simple  proofs  of  the 
formulas,  not  striving  after  original  demonstrations.  Geo- 
metrical proofs  have  been  added  in  many  cases,  experience 
having  shown  that  the  student  is  assisted  by  them  to  a  clearer 
understanding  of  the  subject. 

The  student  is  expected,  in  technical  institutions,  to  acquire 
facility  in  the  use  of  the  tables.  All  of  the  numerical  exam- 
ples have  been  computed  by  the  author,  with  special  attention 
to  correctness  in  the  last  decimal  place,  and  the  arrangement 
of  the  computations  has  been  carefully  considered.  Five-place 
tables  have  been  adopted,  and  the  angles  in  the  examples  are 
given  to  the  nearest  tenth  of  a  minute,  because  the  instruments 
ordinarily  used  by  engineers  are  read  by  the  vernier  only  to 
the  nearest  minute  of  arc,  while  the  angle  corresponding  to  a 
computed  function  may  be  found  usually  to  the  nearest  tenth 
of  a  minute  by  the  use  of  five-place  tables. 

Credit  is  due  particularly  to  the  works  of  Chauvenet,  Snow- 
ball, Beasley,  Woodhouse,  Newcomb,  and  Todhunter,  although 
many  others  have  been  consulted.  A  number  of  the  illustra- 
tive examples  in  Art.  Ill  were  taken  from  Gillespie's  "Land 
Surveying,"  the  numerical  values  being  assigned  by  the  author 
of  this  work. 

The  author  cannot  hope  that  among  so  many  examples 
there  are  no  errors  ;  he  therefore  requests  those  finding  such 
to  kindly  notify  him. 

Rensselaer  Polytechnic  Institutb, 
Troy,  N.  Y. 

8 


1  ^:>i)45 


GREEK  ALPHABET. 


A,  a, 

B,  A 

r,  y, 

A,  S 

E,  6 

z,  ? 

H,  V 

@,  « 

I,  '. 

K,  K 

A,  \ 

M,  M 


a,  a Alpha 

/3,  p Beta 

7,  7 Gamma 

Delta 

.   - Epsilon 

Zeta 

, Eta 

Theta 

Iota 

Kappa 

........  Lambda 

Mu 


N,  1/ .  .  .  .  • Nu 

H,  ? Xi 

O,  0 Omicron 

n,  TT  ........  Pi 

V,  p Rho 

2,  cr,  9 Sigma 

T,  T Tau 

T,  V Upsilon 

^,  (^ PJd 

X,  X ^A*' 

^,  >/r Psi 

12,  o) Omega 


CONTENTS. 


PART  ONE.     PLANE  AND  ANALYTICAL  TRIGONOMETRY. 

CHAP.    I.       MEA.SURBMKNT   OF   ANGLES:    TRIGONOMETRIC    FUNCTIONS   OF 

Angles  less  than  90°. 

PAOB 

Trigonometry  defined 7 

Directed  lines  ;  angles 7 

Measurement  of  angles 8 

Trigonometric  ratios  in  right  triangles 10 

Tables  of  the  ratios 12 

Ratios  for  30°,  45°,  60°'         .         .         .         .        v 12 

Computation  of  the  ratios  when  one  is  given 13 

Measurement  of  angles  in  the  field 14 

Illustrations  of  the  applications  of  the  ratios 16 

CHAP.  11.      Right  Plane  Triangles. 

Facts  derived  from  geometry 17 

Solution  defined 17 

Formulas  employed  in  the  solution  of  right  triangles 17 

Relations  between  the  sides  and  the  angles  ...*...  18 

Methods  of  solution 19 

Isosceles  triangles 23 

Special  methods 23 

CHAP.  III.     Trigonometric  Functions  of  Ant  Angle. 

Generation  of  angles 30 

General  measure  of  an  angle 31 

Coordinates           ............  53 

General  definitions  of  the  trigonometric  functions 35 

Geometrical  representation  of  the  functions 37 

Changes  in  the  values  of  the  functions 40 

Limiting  values  of  the  functions  .........  42 

Graphical  representation  of  the  functions 42 

Two  angles  correspond  to  any  given  function 43 

CHAP,  IV.      Relations  between  the  Functions  of  One  Angle. 

Relations  between  the  functions  of  one  angle,  and  their  applications .         .  46 

Solution  of  trigonometric  equations  containing  one  an^ile    ....  52 

Functions  of  angles  greater  than  360° 54 

Functions  of  90°  ±  x,  270°  ±  x,  180°  ±  ?/,  300°  -  y,  and  -  //         .         .         .54 

The  trigonometric  tables 60 

Transformations 61 

CHAP.  V.     Relations  between  Functions  of  Several  Angles. 

Functions  oi  x  +  y  and  of  x  —  y 64 

Functions  of  2  X  in  terms  of  functions  of  a; 71 

Functions  of  x  in  terms  of  functions  of  2  ic 72 

Multiple  angles 74 

To  change  the  product  of  functions  into  the  sum  of  functions     ...  74 

To  change  the  sum  of  functions  into  the  product  of  functions     ...  76 

Circular  functions 77 

To  prove  that  tan  aj  >  x  >  sin  oj,  when  a;  <  ^  tt 79 

To  prove  that  sin  x,  tan  x,  and  x  approach  equality  as  x  approaches  zero     .  79 

6 


I 


CONTENTS. 


PAGE 

Development  of  sin  x,  cos  x,  and  tan  x 80 

Computation  of  the  trigonometric  functions 82 

Approximate  assumptions 84 

Transformations 85 

CHAP.  VI.     Trigonometric  Equations. 

Equations  containing  multiple  angles 88 

Special  cases 89 

CHAP.  VII.     Oblique  Plane  Triangles. 

Facts  derived  from  geometry 97 

The  sine  proportion 97 

c^  =  a^  -}-  b"^  -  2 ab cosy 98 

Methods  of  solution .  99 

Metliods  of  solution,  using  right  triangles 107 

Areas 110 

Illustrative  examples 112 

PART  TWO.    SPHERICAL   TRIGONOMETRY. 

CHAP.  VIII.     Definitions  and  Constructions. 

Spherical  trigonometry  defined "       .         .  123 

Representation  of  trihedral  angles        .         . 123 

Limitation  of  values 124 

Definitions,  relations,  and  constructions 124 

The  polar  triangle 126 

Facts  derived  from  geometry 127 

Construction  of  triangles 127 

CHAP.  IX.     General  Formulas. 

cos  a  =  cos  b  cos  c  +  sin  6  sin  c  cos  a 130 

cosa  =— COS/3C0S7  +  sin^sin7Cosa 132 

The  sine  proportion       ...........  133 

Additional  formulas      ...........  134 


CHAP.  X.      Right  Spherical  Triangles. 


Formulas 
Napier's  rules 
Species . 


135 

136 

.138 

Methods  of  solution 139 

Isosceles  triangles         ...........     143 

Quadrantal  triangles 144 

CHAP.  XI.     Oblique  Spherical  Triangles. 

To  find  an  angle,  given  the  three  sides 146 

To  find  a  side,  given  the  three  angles 148 

Napier's  analogies 150 

Gauss's  equations 152 

Species 153 

Methods  of  solution .  155 

Methods  of  solution,  using  right  triangles ',      ' .  168 

CHAP.  XII.     Applications  of  Spherical  Trigonometry. 

Distance  between  two  points  on  the  earth's  surface 

Miscellaneous  applications 

Spherical  excess    . 

Legendre's  theorem 

Astronomical  definitions 

Astronomical  applications 


174 
175 
181 
184 
187 
189 


PART   ONE. 

PLAJ{E  AKB  AJ^ALYTICAL   TRIGOKOMETRY. 


CHAPTER   I. 

MEASUREMENT    OF    ANGLES;     TRIGONOMETRIC    FUNCTIONS    OF 
ANGLES   LESS   THAN  NINETY    DEGREES. 

1.  Analytical  Trigonometry  treats  of  the  relations  of  lines 
and  angles  by  algebraic  methods.  In  Plane  and  Spherical 
Trigonometry,  these  relations  are  applied  to  the  solution  of 
plane  and  spherical  triangles. 

2.  Directed  Lines;  Angles. — A  directed  line  is  one  whose 
beginning,  direction,  and  length  are  known.  The  direction  of 
the  line  is  indicated  by  the  order  of  the  letters  in  its  symbol  \ 
for  instance,  the  line  AB  is  drawn  from  A  to  B.  If  one  direc- 
tion along  the  line  is  considered  positive,  the  opposite  direction 
will  be  negative  ;  thus,  if  the  line  AB  is  positive,  the  line 
BA  will  be  negative,  their  numerical  measures  being  equal,  or 

line  AB  =  —  line  BA. 

An  angle  is  the  figure  formed  by  two  intersecting  lines,  the 
point  of  intersection  being  the  vertex. 

The  angle  between  any  two  given  lines,  whether  intersect- 
ing or  not  intersecting,*  is  defined  to  be  the  same  as  the  angle 
formed  by  two  lines  drawn  through  any  point  parallel  to  and 
in  the  same  direction  as  the  given  lines.  Hence  an  angle  may 
be  defined  as  the  difference  in  direction  of  two  directed  lines. 

*  That  is,  parallel  or  in  space. 
7 


8  PLANE  AND  ANALYTICAL  TRIGONOMETRY. 

3.  Measurement  of  Angles.  —  Two  methods  of  measuring 
angles  are  in  common  use,  —  the  sexagesimal  and  the  circular 
or  natural  methods. 


4.  Sexagesimal  Measure.*  —  The  circumference  of  a  circle 
described  about  the  vertex  of  the  angle  as  a  center  is  divided 
into  360  equal  parts,  and  the  angle  at  the  center  subtended  by 
one  of  these  parts  is  taken  as  the  unit.  The  length  of  one  of 
these  divisions  of  the  circle  will  depend  upon  its  radius  ;  but 
the  corresponding  angle  at  the  center  will  be  independent 
of  the  radius,  since  it  is  -jj^  of  four  right  angles.  This  unit 
angle,  called  a  degree^  is  divided  into  60  parts  called  minutes^ 
each  of  which  is  subdivided  into  60  parts  called  seconds. 
These  are  marked  °,  ',  "  ;  thus  43°  14'  35". 2  is  read,  "43  de- 
grees, 14  minutes,  and  35.2  seconds." 

How  many  degrees  are  there  in 

1.  Two  tbirds  of  four  riglit  angles  ?  Ans.  240°. 

2.  Two  fifths  of  three  right  angles  ?  Ans.  108°. 

3.  Five  sixths  of  two  right  angles  ?  Ans.  150°. 

5.  The  Circular  or  Natural  Measure.  —  From  geometry  we 
know  that  in  any  two  concentric 
circles  the  arcs  intercepted  by  any 
angle  at  the  center  are  to  each  other 
as-  the  radii  of  the  circles.  Therefore, 
if  ACB  be  any  central  angle,  we  have 

arc  ^5     'dvcA'B' 


CA  CA' 


(1) 


Hence  the  length  of  the  intercepted 
arc  divided  by  the  radius  is  a  number 
that  is  always  the  same  for  the  same 
angle,  no  matter  what  the  radius  may  be. 

We  also  know  that  in  any  circle  any  two  central  angles  are 
to  each  other  as  their  intercepted  arcs,  and  therefore  as  the 
quotients  of  their  intercepted  arcs  divided  by  the  radius.  We 
can,  then,  use  these  quotients  to  measure  the  angles. 

V 

*  From  sezagesimus,  sixtieth. 


MEASUREMENT  OF   ANGLES.  9 

The  circular  measure  of  an  angle  is  the  quotient  obtained 
by  dividing  the  length  of  its  intercepted  arc,  in  a  circle  whose 
center  is  at  the  vertex  of  the  angle,  by  the  radius  of  the  circle. 
Thus,  if  c  is  the  circular  measure  of  the  angle,  I  its  intercepted 
arc,  and  r  the  radius,  we  have 

«=-  (2) 

If  the  radius  of  the  circle  is  unity, 

c  =  «.  (3) 

Hence  the  circular  measure  is  represented  by  the  length  of  the 
intercepted  arc  in  the  circle  whose  radius  is  unity. 

The  angle  whose  circular  measure  is  one,  that  is,  whose 
intercepted  arc  is  equal  to  the  radius,  is  called  the  radian. 

1.  The  length  of  the  intercepted  arc  of  a  central  angle  is  4  feet  in  a  circle 
whose  radius  is  2  feet;  the  length  of  the  intercepted  arc  of  another  central 
angle  is  20  meters  in  a  circle  whose  radius  is  5  meters.  Show  that  the  second 
angle  is  twice  as  large  as  the  first. 

2.  In  a  circle  with  a  radius  of  10  inches,  the  intercepted  arc  of  a  central 
angle  is  5  inches,  and  that  of  an  angle  whose  vertex  is  on  the  circumference  is 
10  inches.    Find  their  circular  measures.  Ans.  \. 

6.  Relation  between  the  Two  Measures.  —  Two  right  angles 
are  measured  by  180°,  and  also  by  7rr*  ^  r  =  tt,  since  irr  is  the 
semicircumference  of  a  circle  whose  radius  is  r.  Hence,  using 
the  equality  sign  to  represent  "  corresponds  to,"  we  have 

180°  =  1x1^11  circular  measure  5  (1) 

.  •.  1°  =  -^  in  circular  measure.  (2) 

180 

Again,         tt  in  circular  measure  =  180° ;  (3) 

.  ••  1  in  circular  measure  =  i^^.  C4) 

.-.  1  in  circular  measure  =  57°. 29577  95  +  (5) 

.-.  1  in  circular  measure  =  206  264''.806.  (6) 


1.  What  is  the  circular  measure  of  120°  ?  Ans.  120  x  -^  =  I 

2.  What  is  the  circular  measure  of  10°  10'  10"  ? 


180      ' 


The  circular  measure  of  1°  is  -— ,  and  that  of  1"  is But 

180  180  X  60  X  60 

10°  10'  10"  =  36610".      .  •.  Circular  measure  of  10°  10'  10"  =       ^^^^^  "^ 


180  X  QO  X  60 


♦  IT  denotes  the  ratio  of  the  circumference  of  a  circle  to  its  diameter,  and  is 
the  number  3.14159  265+. 


10  PLANE   AND  ANALYTICAL   TRIGONOMETRY. 

3.  What  is  the  sexagesimal  measure  of  the  angle  whose  circular  measure 
^^^'^^  w  =  180°;     .-.  ^7r=60°. 

4.  What  is  the  sexagesimal  measure  of  the  angle  whose  circular  measure 

^^^        TT  ..    •      •      1                         180°            ,                   -    ,    120° 
Unity  in  circular  measure  = ;    .  •.  f  corresponds  to 

TT  IT 

6.    What  are  the  sexagesimal  and  circular  measures  corresponding  to  f  of 
three  right  angles  ?  Ans.  60° ;  \  ir. 

6.    The  sexagesimal  measures  of  two   angles  are  22°  30'  and  43°  14'  3". 

1350  IT  155643  tt 

Show  that  their  circular  measures  are  and 


180  X  60  180  X  60  X  60 

7.  The  circular  measures  of  three  angles  are  y^j  tt,  |  tt,  and  J^  v.  Show 
that  their  sexagesimal  measures  are  15°,  40°,  and  3°  36'. 

8.  The  circular  measures  of  three  angles  are  i,  f ,  and  |.     Show  that  their 

.      ,                         45°   300°        ,  40° 
sexagesimal  measures  are  — , ,  and  _ . 

TT  IT  TT 

9.  Find  the  sexagesimal  and  circular  measures  corresponding  to 

(a)  Seven  tenths  of  four  right  angles.  Ans.  252°  ;  |ir. 

(6)  Five  fourths  of  two  right  angles.  Ans.  225°  ;  |  t. 

(c)   Two  thirds  of  one  right  angle.  Ans.  60°  ;  |  tt. 

7.  Centesimal  Measure.  —  In  this  system,  proposed  by  the 
French,  the  right  angle  is  divided  into  100  parts  called  grades, 
each  of  which  is  subdivided  into  100  parts  called  minutes, 
each  minute  being  divided  into  100  parts  called  seconds; 
marked  ^    \   '\ 

8.  Trigonometric  Ratios.  —  Let  the  sides  of  a  right-angled 
triangle  be  denoted  as  shown  in  Fig.  2.  The  trigonometric 
ratios  may  be  defined  as  follows: 

The  sine  of  an  angle  =  -^ P- ;  written     sin  -4  =  - 

hypotenuse  h 

n^^  .  o  I        side  adiacent         .^^  .a 

ihe  cosine       oi  an  angle  =  -.; ~ ;  written     cos  A  =  ~ 

nypotenuse  h 

The  tangent     of  an  angle  =  -r-^ ^ -;  written    tan  A  =  - 

side  adjacent  « 

The  cotangent  of  an  angle  =  ^^ — t—  :  written     cot  ^  =  - 

^  side  opposite  o 

The  secant       of  an  angle  =   .  ^^   ,. -;  written     sec^  =  - 

side  adjacent  « 

The  cosecant    of  an  angle  =  -^^ r— ;  written  cosec  A  =  - 

side  opposite  o 

These  fundamental  equations  should  be  thoroughly  memorized. 


O) 


MEASUREMENT  OF   ANGLES. 


11 


Fig.  2. 


Fig.  8. 


9.  The  Ratios  are  Constant  for  Any  One  Angle.  —  In  Fig.  3 
let  BAO  and  BAF  be  two  angles  differing  by  a  quantity  as 
sin^ll  as  Ave  please.  At  any  two  points  B  and  I)  on  AB,  draw 
BF  and  BG-  perpendicular  to  AB ;  with  ^  as  a  center,  and 
radius  A  (7,  describe  the  arc  OIT,  and  draw  Lff  perpendicular 
to  AB.     The  triangles  BAO  and  BAF  are  similar. 


BC^BF 
AC     AF 

BO     BF 


a  constant 


sin  a;. 


=  ----  =  a  constant 


AB     AB 


side  opposite 
hypotenuse 

side  opposite      , 

— — =  tan  Xm 

side  adjacent 


AC     AF  ,      ,        hypotenuse 

— -—  =  — —  =  a  constant  =    ,•:       -, . =  sec  x, 

AB     AB  side  adjacent 

10.    The  Values  of  the  Ratios  differ  for  Different  Angles.  — 

From  Fig.  3  we  have,  since  AH  =  AO^ 

BO       .     •  LH     LH 

sin.  =  —   andsin^  =  -^=  — ; 

BO       . ^  BF 

tan  X  =  — — ;  and  tan  y  = ; 

AB  ^      AB 

AO       .  AF 

sec  :c  =  —  and  sec  ^  =  — . 


11.    The  Angle  may  he  constructed 
when  One  of  the  Ratios  is  known.  — 

Let  sin  x  =  ^.  With  any  convenient 
radius  AO,  describe  a  circle  about  A 
as  a  center.  Draw  AB  perpendic- 
ular to  AB,  and  on  it  lay  off  AB= 
^AO;  draw  i> (7  parallel  to  AB  until 


FiQ.  4. 


12  PLANE   AND  ANALYTICAL   TRIGONOMETRY. 

it  intersects  the  circle  at  0 ;  join  A  and  (7,  and  BA  C  will  be 
the  required  angle,  since 

.     ^,ri     BC     AD     1 

sin  BA  C  =  -— -  =  -TT^  =  - . 
AC     AC     2 

Let  tan  a;  =  J.  Lay  off  any  convenient  distance  AB  ;  at  B 
draw  BC  perpendicular  to  AB,  and  lay  off  J9(7=  ^  AB ;  join 
A  and  (7,  and  ^J.  C  will  be  the  required  angle,  since 

tani?^(7=4^=f. 

Let  sec  x  =  2.  Lay  off  any  convenient  distance  AB  ;  erect 
the  perpendicular  line  BC;  with  a  radius  J. (7=  2  J.^  describe 
an  arc  cutting  BC  at  C;  join  ^  and  C,  and  jB^Cwill  be  the 
required  angle,  since 

secj&A(7=4^=2. 

Let  the  student  construct  the  angle  whose  cosine  is  |,  the  angle  whose 
cotangent  is  5,  and  the  angle  whose  cosecant  is  4. 

12.  We  therefore  conclude  that  to  any  one  angle  there 
will  correspond  a  special  value  of  each  of  these  ratios,  that  the 
value  of  each  ratio  will  differ  for  different  angles,  and  that,  if 
any  one  of  these  ratios  is  given,  the  angle  may  be  constructed. 

13.  Tables  of  Sines,  Cosin*es,  etc.  —  The  values  of  these  ratios 
for  angles  between  0°  and  90°  have  been  computed,  and  are 
given  in  tables  so  arranged  that  the  values  corresponding  to 
any  angle  may  be  readily  found.  The  tables  of  natural  sines., 
etc.,  contain  the  actual  values  of  these  ratios  ;  while  the  tables 
of  logarithmic  sines,  etc.,  contain  their  logarithms. 

14.  Ratios  for  30%  45°,  60°. 

(rt)  Ratios  for  45°.  —  In  Fig.  5  let  the  angle  A  =  45°  ;  then 
J5=90°-^  =  45°. 

.-.  AC  —  CB,  since  they  are  opposite  equal  angles. 

Let  AC=  a;  then  CB  =  a^  and  AB  =  Va^  +  a^  =  a  V2. 


MEASUREMENT  OF  ANGLES. 


13 


^°^'''°=3l=^'  cot45'=^|=l;  cosec46o=^=V-2. 


(5)  Ratios  for  30°  and  60°.  —  In  the  equilateral  triangle 
ABC  (Fig.  6),  let  .A.S  =  a  ;  draw  D^  perpendicular  to  AO ; 
AC  will  be  bisected  at  i>,  making  J.i>  =  Ja,  and  the  angle 
^^i)=angle  i>^(7=30°. 

Also  DB  =  Va2  -  1  ^2  =  1  a  Vs. 


sin  ABi>  =  sin  30^^ 


^^     2 


tan  30°  = 


cos  30°  =  ^=—;    cot  30° 
AB      2 

DB     V3 


DB     V3 
^  =  V3 


siriZU^=sin60°  =  :^=^;    tan60°  =  — =  >/3 
^^2  ^Z> 


2)^ 
AD 


cos60°  =  — =  i;      *cot60°  = 

^^2  Z)^     V3 

Note  that  the  sines  of   30°,  45°,  and  60°,  are 
spectively. 


sec  30°  =  ^  =— 
DB     V3 

cosec  30°  =  ^  =  2. 


AD 

sec  60°  =  :^^  =  2  : 

cosec  60°  =  :^=  A. 
DB     V3 

^(v/l,/^V2,  and^VS   re- 


15.    The  Ratios  are  not  Independent  of  Each  Other ;  for  we 

have  from  Fig;.  2, 

so  that  if  two  of  the  three  quantities  A,  o,  and  a,  are  given,  the 
third  can  be  found.  Hence  if  we  know  one  of  the  ratios,  that 
is,  the  relative  values  of  two  of  the  three  elements,  we  can  de- 
termine the  relative  value  of  the  third  element,  and  from  it 
the  other  ratios. 


14 


PLANE   AND  ANALYTICAL  TRIGONOMETRY. 


Thus  if  tan  a:  =  |,  and  the  other  ratios  are  required,  we  have 


tana: 


;   let  0  =  3,  a  =  4  ;  then  A  =  5. 


a 

4' 

' 

••• 

sin  2:  = 

0 

3 
5' 

COS  X 

a 

4 
^5' 

cot  2: 

0 

4 

^3' 

sec  2: 

_h_ 

a 

5 

cosec  X  - 

0 

5 
3* 

Having 

given  the  ratic 

on 

the  left, 

find  the  ratios  on  the 

right : 

sin  a;. 

cos  a?. 

tan  a. 

cot  85. 

seca;. 

cosec  85. 

o.-^  ^       8 

15 

8 

15 

17 

17 

1. 

sin  X  =  — 

— 

17 

17 

15 

8 

15 

8 

5 

12 

12 

5 

13 

13 

2. 

cos  x  =  — 

— 

13 

13 

5 

12 

5 

12 

8. 

tan  a;  =  — 

7 

24 

24 

25 

25 

24 

25 

25 

7 

24 

7 

4. 

cot  a  =  2 

1 

V5 

2 

V5 

1 
2 

,  — 

IVB 

V5 

29 

21 

20 

21 

20 

29 

b. 

seca;  =  — 

20 

29 

29 

20 

21 

21 

6. 

cosec  x  =  3 

1 
3 

1--. 

Iv^ 

2V2 

\^^ 

— 

16.  Measurement  of  Angles  in  the  Field.  —  In  Fig.  7, 
FG-HK  represents  a  fixed  graduated  circle,  and  ABDE  a 
circle  resting  on  the  plate  FGrHK^  and  capable  of  moving 
about  a  pivot  at  O';  J  and  0  are  two  small  rods  fixed  to 
ABBE^  and  perpendicular  to~the  planes 
of  the  circles  ;  and  il[f  is  a  mark  on  the 
circle  ABBE  in  the  same  line  with  /, 
(7,  and  0.  If  we  wish  to  measure  the 
horizontal  angle  between  two  distant 
objects,  two  church  towers,  for  ex- 
ample, we  proceed  as  follows :  first 
place  the  circles  in  a  horizontal  posi- 
tion ;  revolve  the  circle  ABBE,  look- 
ing along  the  line  10,  until  the  line  of 
sight  passes  through  one  of  the  objects,  and  note  the  reading 


MEASUREMENT  OF  ANGLES. 


16 


of  the  circle  opposite  the  mark  M;  then  revolve  the  circle 
ABBE,  being  careful  not  to  move  FGHK,  until  the  line  of 
sight  passes  through  the  second  object,  and  note  the  new  read- 
ing of  the  circle  opposite  the  mark  M.  The  difference  between 
the  two  readings  will  be  the  angular  distance  required. 

17.  The  Engineers'  Transit,  shown  in  Fig.  8,  is  used  in 
measuring  horizontal  and  vertical  angles.  The  lower  circle  is 
provided  with  two  levels,  by  which  its  horizontality  is  tested. 


Fig.  8. 


The  rods  I  and  0  are  replaced  by  the  telescope  with  a  system 
of  intersecting  wires  in  the  common  focus  of  the  object  glass 
and  eyepiece,  the  telescope  being  capable  of  rotation  about 
an  axis  parallel  to  the  horizontal  circle.  The  circle  fixed  to 
the  axis  of  the  telescope  is  vertical  when  the  plate  bearing  the 
upright  supports  is  horizontal. 


16 


PLANE   AND  ANALYTICAL   TRIGONOMETRY. 


18.    Illustrations  of  the  Application  of  the  Ratios.* 

1.  A  rope  fastened  to  the  top  of  a  vertical  pole  60  feet 
high,  and  to  a  stake  driven  in  the  ground,  is  inclined  at  an 
angle  of  30°.     How  far  is  the  stake  from  the  bottom  of  the 

pole  ?     How  long  is  the  rope  ? 

-^  =  tan  30°  =  —. 
AC  V3 

•••  ^^  =  V3a^  =  60V3feet. 


Fig.  9. 


OB        .     ono      1 
— —  =  sm  30°  =  -. 
AB  2 

.-.  ^j5  =  2  (7^  =  120  feet. 


2.    The  angle  at  the  vertex  of  a  right  circular  cone  is  60°, 
and  the  slant  height  is  10  inches.     What  is  the  altitude  and 

the  radius  of  the  base  of  the  cone  ? 


Fig.  11. 


OB       _„  ono       V3 

_  =  cos30  =— . 


OB  =  ^AB 


5  V3  inches. 


^=sin  30^  =  1 

. •.  AC=  ^  AB  =  5  inches. 

3.  The  top  of  a  ladder  30  feet 
long  rests  on  the  upper  edge  of  a 
wall  15  feet  high.  What  is  the 
inclination  of  the  ladder  ? 


sin  CAB  = 


05^15^1. 

AB     30     2' 


but     sin30°  =  V.     .'.  (7^^  =  30°. 


In  these  cases  the  ratios  corresponding  to  the  angles  were 
known  from  Art.  14.  Usually  it  will  be  necessary  to  refer 
to  the  tables  in  solving  problems  involving  the  ratios. 


It  is  assumed  that  the  ground  is  horizontal. 


CHAPTER   II. 

RIGHT  TLANE  TRIANGLES. 

19.  It  has  been  shown  in  Geometry  that  a  right-angled  tri- 
angle can  be  constructed  when  two  elements  *  besides  the  right 
angle  are  known,  one  of  the  known  elements  being  a  side.  We 
also  know  that 

(1)  The  hypotenuse  is   greater  than   either  of   the  other  ' 
two  sides. 

(2)  The  hypotenuse  is  less  than  the  sum  of  the  other  two  -^ 
sides. 

(3)  The  sum  of  the  two  acute  angles  must  be  90°.    '^ 

(4)  The  greater  side  is  opposite  the  greater  angle.     ' 

(5)  The  square  on  the  hypotenuse  is  equal  to  the  sum  of  "^ 
the  squares  on  the  other  two  sides. 

20.  A  triangle  is  said  to  be  solved  when,  having  some  of  the 
elements  given,  the  others  have  been  found  by  some  process. 

21.  The  Solution  of  a  Right  Triangle  is  effected  by  means  of 
the  trigonometric  ratios.     Each  equa- 
tion,    as     sin  ^  =  -,    contains     three 

c 

quantities ;  and  two  of  them  must  be 
known  in  order  that  the  third  may 
be  found.  Hence  in  any  particular 
case  we  use  the  equations  that  con- 
tain the  two  given  elements ;  thus,  if  ^^ 
a  and  h  are  given,  we  use  tan  J.  =y  via-^ii 

0 

to  find  A,  and  then  e  may  be  found  from  either  sin  A  =-  or 

COS  A  =  -, 
c 

*  The  elements  of  a  triangle  are  the  three  sides  and  the  three  angles. 

CROCK.    TRIG.  —  2  17 


18  PLANE  AND  ANALYTICAL   TRIGONOMETRY. 

The  equations  used  in  the  solution  of  right  triangles  are 


Fia.  18 


sin  A  =  -  =  cos  B, 
c 

cos  A=  -  =  sin  B, 

c 

tan  J.  =  -  =  cot-B. 

0 

cot  A  =  -  —  tan  B. 
a 

A  +  B  =  90°. 

^  =  ^2  +  ^2. 


(1) 


22.    From  the  Trigonometric  Ratios  we  have 


tanJ.=     ;    .*.  a  =  5tanJ., 

0 


cot  B  =  -;    .'.  a  =  b  cot  B^ 

0 


(1) 


or,  any  side  of  a  right  triangle  is  equal  to  the  other  side  multiplied 
by  the  tangent  of  the  angle  opposite^  or  by  the  cotangent  of  the 
angle  adjacent^  to  the  side  itself. 


sin^  =  — ;    .*.  a  =  (?  sin  ^, 

c 

eosB  =  -;    .*.  a  =  c  cos B, 

c 


(2) 


or,  any  side  is  equal  to  the  hypotenuse  multiplied  by  the  sine  of 
the  opposite  angle^  or  by  the  cosine  of  the  adjacent  angle. 


sec  J.  =  -; 


'=h  sec^, 


cosec^=-;     .-.  c=5cosecJ5, 
b 


(3) 


or,  the  hypotenuse  is  equal  to  a  side  multiplied  by  the  secant  of 
the  adjacent  angle^  or  by  the  cosecant  of  the  opposite  angle. 

Note.  — The  secant  of  an  angle  is  the  reciprocal  of  its  cosine,  and  the  cose- 
cant is  the  reciprocal  of  its  sine  ;  hence  the  logarithm  of  the  secant  is  the  arith- 
metical complement  of  that  of  the  cosine,  and  the  logarithm  of  the  cosecant  is 
the  A.  C.  of  that  of  the  sine,  or 


log  sec  X  =  colog  cos  x,  and  log  cosec  x  =  colog  sin  x. 


RIGHT  PLANE  TRIANGLES. 
23.    Case  I.   Given  c  and  A, 


Formulas: 


a  =  (?  sin  A, 
h  =  c  cos  A, 
^=90° -A 


19 


1.   Solve  the  triangle  when  c  =  1.0034,  and  A  =  42^  lO'.S. 
.-.  B  =  90°  -  ^  =  47°49'.7. 

(a)  By  natural  functions. 

a  =  c  sin  ^  =  1.0034  x  0.67136  =  0.67364. 
6  =  ccos^  =  1.0034  X  0.74114  =  0.74366. 

(b)  By  the  use  of  logarithms. 

a  =  c  sin  ^  ;   .  •.  log  a  =  log  c  +  log  sin  A. 
6  =  c  cos  ^  ;  .  •.  log  6  =  log  c  +  log  cos  A. 

Always  write  first  all  the  formulas  that  will  be  used  in  the 
problem;  then  write  them  in  a  form  adapted  to  logarithmic 
computation ;  then  refer  to  the  tables  and  write  the  logarithms 
in  their  proper  places.  Thus  in  this  case  we  arrange  the  work 
as  follows: 

logc=  logc  = 

+  log  sin  ^  =  +  log  cos  A  = 

.-.  loga=  .-.  log 6  = 

.-.  a=  .'.  6  = 


The  positive  signs  preceding  log  sin  A  and  log  cos  A  indicate 
that  they  are  to  be  added  to  log  c. 

We  now  find  the  angle  A  in  the  table  of  logarithmic  func- 
tions and  take  from  the  table  both  log  sin  A  and  log  cos  A, 
writing  them  in  their  proper  places.  Then  we  refer  to  the 
table  of  logarithms  of  numbers  and  find  log  c,  writing  it  oppo- 
site log  c.  Then  we  add  the  proper  quantities  to  find  log  a  and 
log  5,  finally  looking  in  the  table  of  the  logarithms  of  numbers 
for  the  numbers  corresponding  to  the  computed  values  of  log  a 
and  log  b. 

The  arrangement  on  the  right  is  preferable,  since  it  saves 


log  sin  A  = 

(1) 

logc  = 

(3) 

log  cos  A  = 

(2) 

.-.  loga  =  (l)  +  (3) 

(4) 

a  = 

(6) 

.'.  log  6  =(2)4- (3) 

(5) 

b  = 

(7) 

20 


PLANE  AND  ANALYTICAL   TRIGONOMETRY. 


the  writing  of  one  line.     The  numbers  in  the  parentheses  indi- 
cate the  order  in  which  the  quantities  should  be  found. 


0.00147 
9.82695 


logc: 
10    +  log  cos  ^ 


log6  =  9.87137 
6  =  0.74365 


0.00147         or  log  sin  A  =  9.82G95  -  10 
9.86990-10  logc  =  0.00147 

log  cos  ^  =  9.86990  -  10 


10 


Check : 

c  +  6  =  1.74705 
c-b  =  0.25975 


b) 


loga  =  9.82842  -  10 

a  =  0.67363 
log6  =  9.87137 -10 
6  =  0.74365 


log  (c+ 6)  =  0.24230 

log  (c  -  &)  =  9.41456  ~  10 


.-.  loga2  =  9.65686 
loga  =  9.82843 


Exact  agreement  is  not  expected,  since  the  tables  give  the  values  of  the 
functions  only  to  the  nearest  unit  in  the  fifth  decimal  place.  The  —  lO  is  usually 
omitted,  and  sin  A  is  written  for  log  sin  J.,  when  there  is  no  danger  of  confusion. 

2.  Solve  the  triangle  when  c  =  34.687,  and  B  =  49°  8'.4. 

Ans.  A  =  40°  51'.6  ;  b  =  26.234  ;  a  =  22.6925. 

3.  Solve  the  triangle  when  c  =  305,  and  A  =  63°  31'.14,  using  the  natural 
functions.  ^^^^  ^  ^  273.00  ;  h  =  136.00. 


4.    Solve  the  triangle  when  c  =  205,  and  B  =  49°33'.01,  using  the  natural 

Ans.  a  =  133.00  ;  b  =  156.00. 


functions. 


24.    Case  II.     Given  c  and  a. 


Formulas 


sin  A  —  -. 

c 

b  =  a  cot  A  =  c  cos  A. 


1.    Solve  the  triangle  when  c  =  8.7982,  and  a  =  3.1292. 

.*.  logsin^  =  loga  —  logc;  logft  =  loga  +  logcot^  =  logc  +  logcos  A 

log  a  =  0.49544  log  a  =  0.49544  log  c  =  0.94439 

-logc  =  0.94439  +logcot^  =  0.41958         +logcos^  =  9.97063  -  10 


log  sin  ^  =  9.55105 - 
A  =  20°  50M 
^  =  69°   9'.9 


10 


log  6  =  0.91502 
b  =  8.2228 


log6  =  0.91502 
b  =  8.2228 


RIGHT  PLANE  TRIANGLES. 


21 


log  cot  ^  =  0.41958  (5) 

loga  =  0.49544  (1) 

-logc  =  0.94439  (2) 

log  cos  ^  =  9.97063  (6) 

logsin^  =  9.55105    (l)-(2) 
A  =  20°  50'.1  (4) 
i?  =  69°    9'.9 

(l)  +  (5) 
(2) +  (6) 
b  =  8.2228 


Check  :  b^=(c-  a){c-\-  a) 
c-a=    5.6690,  log  (c  -  a) 
c-ha  =  11.9274 


0.75361 

log  (c  + a)  =1.07656 
log  62 


1.83006 
log  6  =0.91603 


log6  =  0.91602     { 


2.  Solve  the  triangle  when  c  =  369.27,  and  b  =  235.64. 

Ans.  A  =  50°  20'.9  ;  5  =  39°  39M  ;  a  =  284.31. 

3.  Solve  the  triangle  when  c  =  281,  and  a  =  160,  using  the  natural  functions. 

Ans.  A  =  34°  42'.5  ;  b  =  231.00  or  231.01. 

4.  Solve  the  triangle  when  c  =  365,  and  b  =  76,  using  the  natural  functions. 

Ans.  A  =  77°  58'.93 ;  a  =  357.00. 


25.    Case  III.     Given  a  and  b. 


Formulas: 


tan^  = 


c  = 


sin  A      cos  A 
5  =  90°  -  A. 


1.  Solve  the  triangle  when  a  =  169.03,  and  b  =  203.44. 

.  •.  log  tan  ^  =  log  a  -  log  6  ;  log  c  =  log  a  -  logsin  A  =  \ogb  —  log  cos^. 

log  a  =  2.22796 
-log  6  =  2.30843 

log  tan  A 


log  a  =  2.22796 
log  sin  ^  =  9. 80555 


10 


log  6  =  2.30843 
log  cos  A  =  9.88602  -  10 


9.91953-10 
^  =  39°43'.3 
J5  =  50°16'.7 


logc  =  2.42241 
c  =  264.49 


logc  =  2.42241 
c  =  264.49 


or 


*loga  =  2.22796 

(1) 

logsin^  =  9.80555 

(5) 

log  cos  ^  =  9. 88602 

(6) 

log6  =  2.30843 

(2) 

log  tan  ^  =  9.91953 

(3) 

al  =  39°43'.3 

(4) 

J5  =  50°16'.7 

logc  =  2.42241 
c  =  264.49 

1(2)- 

-(5) 
-(6) 

Check :  a2  =  c2  -  62 

c  +  6  =  467.93 
c-6=    61.06 

log(c  + 6)=  2.67018 
log(c-  6)=  1.78569 

.-.  loga^  =4.45587 
log  a  =2.22794 


*  This  form  is  preferable. 


22 


PLANE   AND  ANALYTICAL   TRIGONOMETRY. 


2.  Solve  the  triangle  when  a  =  4.8199,  and  b  =  2.6492. 

Ans.  A  =  61°  12'.8  ,  B  =  28"  il'.l ;  c  =  5.4999. 

3.  Solve  the  triangle  when  a  =  60,  and  6  =  91,  using  the  natural  functions. 

Ans.  A  =  33°  23'.9  ;  c  =  109.00. 

4.  Solve  the  triangle  when  a  =  72,  and  b  =  65,  using  the  natural  functions. 


Ans.  A  =  47°  55'.5 ;  c  =  97.000. 
B 


26.    Case  IV.     Given  a  and  A, 

h  =  a  cot  A. 


Formulas: 


c  = 


B 


sin  A     cos  A 
90°  -  A, 


1.    Solve  the  triangle  when  a  =  613.35,  and  A  =  40°  12'.6. 
.-.  5  =  90° -^=49° 47 '.4. 

log  &  =  log  a  +  log  cot  A. 

log  c  =  log  a  —  log  sin  A  =  log  b  —  log  cos  A. 


loga  =  2.78770 
cot^  =  0.07295 


log&  =  2.86065 
b  =  725.52 


loga  =  2.78770 
logsin^  =  9.80996 -10 

logc  =  2.97774 
c  =  950.04 


log6  =  2.86065 
log  cos^  =  9.88291  -  10 

logc  =  2.97774 
c  =  950.04 


or    log  sin  ^  =  9.80996  (1) 

log  a  =  2.78770  (3) 

log  cot  ^  =  0.07295  (2) 

logc  =  2.97774  (3)-(l) 

c  =  950.04 

log6  =  2.86065  (3) +  (2) 

b  =  725.52 


Check:  a2=(c  +  &)(c-fe) 

c  +  b  =  1675.56,   log  (c  +  b)=  3.22416 
c-b=   224.52,    log  (c  -  6)  =  2.35126 

log«''^  =  5.57542 
loga  =  2.78771 


2.  Solve  the  triangle  when  a  =  3.6378,  and  B  =  69°  23'. 5. 

Ans.  A  =  20°  36'.5  ;  b  =  9.6738  ;  c  =  10.335. 

3.  Solve  the  triangle  when  b  =  160,  and  A  =  55°  17'.48,  using  the  natural 

functions. 

Ans.  c  =  281.00  ;  a  =  231.00. 

4.  Solve  the  triangle  when  a  =  340,  and  A  =  60°  55'.  85,  using  the  natural 

functions. 

Ans.  c  =  389.00  ;  b  =  189.00. 


RIGHT  PLANE   TRIANGLES. 


23 


27.  Isosceles  Triangles. — If  a  perpendicular  to  the  base  is 
drawn  from  the  vertex,  it  will  bisect  the  base  and  the  angle  at 
the  vertex,  forming  two  equal  right  tri- 
angles. 

ZABI)=ZI)BC=l^;  AB  =  BO; 

1.    Solve    the    triangle   when    h  =  2.1452,    and 
j8  =  121°  14'.6. 

.-.  ^i?=  1.0726;  ^i52)  =  60°37'.3; 

o  =  90°-^)8=29°22'.7. 

^"^sinT^'     •■•  loga  =  logi6-logsinJ)8. 


log  ^6  ==0.03044 
log  sin  i)a  =  9.94022  -10 

loga  =  0.09022 
a  =  1.2309 


\ogp  =  log  J  6  +  log  cot  J  0. 
log^&  =  0.03044 
+  log  cot  ^  /8  =  9. 75049  -  10 

logp  =  9.78093-  10 
p  =  0.60385 


2.    Solve  the  triangle  when  a  =  52°  10'. 2,  and  a  =  600.2. 

Ans.  /8  =  75°39'.6;  ^6=368.12;   ;>  =  474.07. 

28.    Given  c  and  b  (Special  Method).  — When  b  nearly  equals 

c,  the  angle  found  from  the  formula  cos  J.  =  -  is  uncertain,  the 

c 
tabular  difference  for  the  cosine 

being  so  small  that  a  small  error 
in  cos^  would  produce  a  large 
error  in  A. 

In  the  figure,  AB  bisects  the 
angle  A^  and  BU  is  perpen- 
dicular to  AB;  .'.  BE=  CJ). 
Let  OB  =  x=BB; 


.-.   tan4a  =  - 
2         b 

Also,  CB=a=  CB  +  BB=  OB-^  BUsec  a 

a 


(1) 


a  =  X  -{-  X  sec  a  ; 
a  ab 


X  = 


1  H-  sec  a ' 


X 


1  + 


+  4' 


(2) 


24 


PLANE   AND   ANALYTICAL   TRIGONOMETRY. 


From  (1)  and  (2), 


tan  -|-  u 


c  -^  b         c 


h)(c-b}. 


.  •.  tan  J 


+  6         ^l       ic  +  bf       ■ 

^  C  -\-  0 


Suppose  that  we  wish  to  find  the  greatest  distance  at  sea  at  which  a  moun- 
tain 4.3  miles  high  can  be  seen,  the  earth  being  considered  as  a  sphere  witli 
a  radius  of  3963.3  miles,  and  the  distance  being  measured  as  a 
-A  chord. 

Let  5^=4.3,  and  CB=CB  =  'dmS.S  ;  BD  being  the  distance 

required.    Then  cos  DCA  =  — ,  giving  log  cos  DCA  =  9.9m52  ; 

CA 
and  DCA  as  found  from  the  tables  might  have  any  value 
between  2°  40'.5  and  2°  42'.5. 
Using  (3),  we  have 

CA-CD=       4.3;  log  =  0.63347 
CA  +  CD  =  7930.9  ;   log  =  3.89932 

2)6.73415  -  10 
log  tan  IDCA  =  8.36708  -  10 
bpl.  r=  3.53620 
log  (I  Z>C^)'=  1.90328 

.-.  I  DCA  =  80'm5;     .-.  DCA  =2°  iO'.Ol. 

Then  BD  =  2CDsmhDCA  will  give  the  chord  BD.      The  arc  BD  is  found 
from  the  proportion : 

360°  :  DCA  =  2Tr  X  3963.3  :  arc  BD. 

Note.  —  Eq.  (3)  follows  directly  from  (4),  Art.  69  : 


tan  i  a  =  \ 52ifi;    where  cos  a 

^  ^1+COSa' 


29.  Given  a  and  b  (Special  Method).  — 
^  When  a  and  b  are  nearly  equal,  the  angle 
a  may  be  determined  more  accurately,  as 
follows  : 

Draw  AI),  making  CAD  =  45°,  and 
DU  perpendicular  to  AB.     Then 


tan  BAU  =  tan(a  -  4o°)  = 


AH 


But  DU  =  DB  cos  a  =  COB-  CD')  cos  a 


(a  —  5)  cos  a 


(a  -  b)b 


RIGHT  PLANE   TRIANGLES.  26 

and 

AE  =  AB  —  EB  =  AB  —  BB  sin  a  =  c  —  ^ ^-  = — — 

c  c 

^b^  +  ab 

c 

BE  ^(a-b)b  ^a-b 

AE        ab  +  b'^       a  +  h 

.-.  tan(«-45°)  =  ^.  (1) 

a  -{•  0 

If  b  were  greater  than  a,  the  formula  would  be 

tan(45°-«)=^^^.  (2) 

f 
Note.  —  Eq.  (1)  may  be  found  from  the  relation  proved  in  Art.  100  : 

a  -  6  _  tan  Ha  -  ^)^  ^j^^-^  i  ^^  ^  ^)=  45°,  and  K^  -  /3)=  a  -  46°. 


a  +  6      tanHa  +  /3) 


EXAMPLES. 


Note.  — The  angle  between  the  line  of  sight  and  a  horizontal  plane  is  called 
an  angle  of  elevation  when  the  point  sighted  on  is  above  the  horizontal  plane, 
and  an  angle  of  depression  when  it  is  below  the  horizontal  plane. 

1.  The  shadow  of  a  vertical  pole  30  feet  high  is  40  feet  long.  Find  the 
elevation  of  the  sun  above  the  horizon.  Ans.  36°  52 '.2. 

2.  The  vertical  central  pole  of  a  circular  tent  is  20  feet  high,  and  its  top  is 
fastened  by  ropes  40  feet  long  to  stakes  set  in  the  ground,  the  ground  being 
horizontal.  How  far  are  the  stakes  from  the  foot  of  the  pole,  and  what  is  the 
inclination  of  the  ropes  to  the  ground  ?  Ans.  34.641  feet ;  30°. 

8.  The  top  of  a  lighthouse  is  200  feet  above  the  sea  level,  and  the  angle  of 
depression  to  a  buoy  is  9°  52 '.8.  Find  the  horizontal  distance  of  the  buoy  from 
the  lighthouse.  Ans.  1148.3  feet. 

4.  The  horizontal  distance  from  a  point  to  the  vertical  wall  of  a  tower  is 
1000  feet,  and  the  angle  of  elevation  of  the  top  is  4°  15'.2.  Find  the  height  of 
the  top  of  the  wall  above  the  point.  Ans.  74.370  feet. 

5.  Two  points  A  and  B  are  on  the  opposite  banks  of  a  stream*.  A  line  AG 
at  right  angles  to  ^  J5  is  measured  300  feet  long,  and  the  angle  ACB  is  found  by 
measurement  to  be  62°  30'.4.     What  is  the  distance  from  Ato  B? 

Ans.  576.45  feet. 

6.  From  the  top  of  a  lighthouse,  150  feet  above  the  sea  level,  the  angle  of 
depression  to  a  buoy  was  12°  10'.2,  and  that  to  the  shore,  measured  in  the  same 
vertical  plane  with  the  buoy,  was  62°  14 '.8.  Find  the  distance  in  feet  of  the  buoy 
from  the  shore.  Ans.  Log.  Tables,  616.60 ;  Nat.  Tables,  616.61 


26  PLANE   AND  ANALYTICAL   TRIGONOMETRY. 

7.  The  angle  of  elevation  to  the  top  of  the  vertical  wall  of  a  tower  is 
20°  10  .4,  and  the  angle  of  depression  to  the  bottom  is  10^  11 '.0,  the  horizontal 
distance  from  the  observer  to  the  wall  being  250  feet.  Find  the  height  of  the 
wall.  Ans.  136.802  feet. 

8.  We  wish  to  make  a  ladder  that  would  reach  from  a  point  20  feet  in  front 
of  a  building  to  the  fourth  story,  a  height  of  45  feet.  Find  the  length  of  the 
ladder  and  the  angle  it  would  make  with  the  ground  in  this  position. 

Ans.  49.244  feet;  66°  2'.2. 

9.  The  ridgepole  of  a  roof  is  15  feet  above  the  center  of  the  garret  floor, 
and  the  garret  is  40  feet  wide.  What  is  the  inclination  of  the  roof  to  a  horizon- 
tal plane  ?  Ans.  36°  52 '.2. 

10.  A  chord  of  a  circle  is  20  feet  long,  and  the  angle  at  the  center  subtended 
by  it  is  46°  43'.6.     Find  the  radius  of  the  circle.  Ans.  25.217  feet. 

11.  The  angle  between  two  lines  is  40°  r2'.4,  and  a  circle  whose  radius  is 
5730  feet  is  tangent  to  both  lines.  Find  the  distance  from  the  point  of  tangency 
to  the  point  of  intersection  of  the  two  lines  when  the  circle  is  in  the  smaller 
angle,  and  when  it  is  in  the  larger  angle  formed  by  producing  one  of  the  lines. 

Ans.   15055  and  2097.2  feet. 

12.  The  legs  of  a  pair  of  dividers  are  set  so  that  the  angle  between  them  is 
80°  24'.4.  What  is  the  distance  between  the  points,  the  legs  being  6  inches 
long?  Ans.  7.7460  inches. 

13.  An  equilateral  triangle  is  circumscribed  about  a  circle  whose  radius  is 
10  inches.     Find  the  perimeter  of  the  triangle.  Ans.  GOVS  inches. 

14.  A  wedge  measures  12  inches  along  the  side,  and  its  base  is  2  inches 
wide.     Find  the  angle  at  its  vertex.  Ans.  9^  33'. 6. 

15.  The  side  of  a  regular  decagon  is  2.4304  feet.  Find  tlie  radii  of  the 
inscribed  and  circumscribed  circles.  Ans.  3.7400  feet;  3.9325  feet. 

16.  The  area  of  a  regular  octagon  is  24  square  feet.  Find  the  radius  of  the 
inscribed  circle  and  the  length  of  one  of  the  sides.     Ans.  2.6912  feet ;  2.2295  feet. 

17.  The  radius  of  the  circumscribing  circle  of  a  regular  dodecagon  is  10 
feet.    Find  the  area  of  the  dodecagon.  Ans.  300.00  square  feet. 

18.  A  cord  is  stretched  around  two  wheels  with  radii  of  7  feet  and  1  foot 
respectively,  and  with  their  centers  12  feet  apart.  Prove  that  the  length  of  the 
cord  is  12\/3  +  10  tt  feet. 

19.  A  cord  is  stretched  around,  and  crossed  between,  two  wheels  whose  radii 
are  5  feet  and  1  foot  respectively,  their  centers  being  12  feet  apart.  Prove  that 
the  length  of  the  cord  is  12\/3  +  87r  feet. 

20.  Find  the  radius  and  the  length  of  an  arc  of  1°  of  the  parallel  of  latitude 
at  a  place  whose  latitude  is  42°  43'.  9,  the  earth  being  regarded  as  a  sphere  whose 
radius  is  3963.3  miles.  Ans.  2911.1  miles;  50.809  miles. 

21.  The  altitude  of  a  right  circular  cone  is  4. 1436  feet,  and  the  angle  at  its 
vertex  is  20°  14'.2.     Find  its  convex  surface.  Ans.  9.7780  square  feet. 


RIGHT  PLANE  TRIANGLES. 


27 


22.  The  altitude  of  a  right  pyramid  with  a  square  base  is  14.463  feet,  and 
the  sides  of  the  base  are  each  4.703C  feet.  Find  its  slant  height,  its  lateral  edge, 
and  the  angle  between  a  face  of  the  pyramid  and  its  base. 

Ans.  14.643  feet;  14.831  feet;  80°  45'. 5. 

23.  The  base  of  a  trapezoid  measured  600.430  feet,  and  the  angles  at  the 
ends  of  the  base  were  found  to  be  62°  14'. 3  and  74°  18'. 6.  Find  the  length  of 
the  other  base,  the  altitude  being  40 

feet.  Ans.  568.138  feet. 

24.  Find  the  length  of  the  perpen- 
dicular from  the  vertex  of  the  right 
angle  of  a  triangle  to  the  hypotenuse, 
the  hypotenuse  being  6.4603  inches 
long,  and  one  of  the  angles  of  the  tri- 
angle being  40°  40'. 4. 

Ans.  3.1934  inches. 

25.  A  street-railway  track  is  10  feet 
from  the  curbstone  (FB  =  HD  =  10),  Fig.  20. 
and  in  passing  a  corner  where  the 

street  is  deflected  through  an  angle  of  60°,  the  rail  must  be  4  feet  from  the  corner 


(GC  =  i).    Find  the  radius  of  the  circular  curve. 


Ans.  0C  = 


20-4v/3 


2-^3 
26.   Before  paying  for  a  pavement,  it  was  necessary  to  find  the  area  shaded 

28750 
in    Fig.    21.      Prove   that    it    is 1-  7500  square  feet,  the  streets  being 

50  feet  wide.  ^ 


Fig.  21. 


27.  In  the  egg-shaped  sewer  (Fig.  22),  C  is  the  center  of  the  arc  ADB  with 
a  radius  a  ;  /  and  J,  of  AF  and  BG  respectively  with  the  radii  3  a  ;  and  IT,  of 
FEG  with  the  radius  |  a.     Prove  that  its  area  is 

a2fj+ ^tan-i  ^  + 9  tan-i?- 3^  =  a2f^^  +  §^tan-i?- 3^  =  4.59413  a2, 
\i5      4  6  4/\8  4  4/ 

where  tan-i  -  is  the  angle  whose  tangent  is  ^ 
3  3 


28 


PLANE   AND  ANALYTICAL   TRIGONOMETRY. 


28.  A  hill  rises  1  foot  vertically  in  a  horizontal  distance  of  30  feet.  What 
is  the  difference  of  elevation  of  two  points  that  are  1000  feet  apart,  the  distance 
being  measured  on  the  ground  ? 

log  tan  a  =  8.62288  -  10 
CpL  T'  =  3.53611 

log  a' =  2.05899 
S'  =  6.46365  -  10 


log  sin  o=:  8. 52264 -10 
log  1000  =  3. 

log  diff.  of  elev.  =  1.52264 
difE.  of  elev.  =  33.315  feet. 

29.    The  horizontal  distance  between  the  two  extreme  positions  of  the  end 
of  a  pendulum  40  inches  long  is  4  inches.     Through  what  angle  does  it  swing  ? 


Half-angle  =2°  51 '.96. 


Ans.  5°43'.92. 


30.  The  angular  diameter  of  the  moon 
is  31'.12,  and  its  distance  is  238840  miles. 
Find  its  diameter  in  miles. 


BAD  =  SV. 12, 


and  ^C  =  238  840. 
Ans.  2162.0  miles. 


31.  The  equatorial  horizontal  parallax 
of  the  sun  is  8".  8,  and  the  radius  of  the 
earth  is  3963.3  miles.  Find  the  distance 
of  the  sun  from  the  earth. 


BAC  =  8". 8,  and  BC  =  3963.3. 


Ans.  92  896  000  miles. 


32.    A  circular  chimney  100  feet  high  is  10  feet  in  diameter  at  the  base,  and 
8  feet  at  the  top.     Find  the  angle  at  the  vertex  of  the  cone  of  which  it  is  a 

frustum. 

Half-angle  =  34'.376.  Ans.  1°8'.752. 


Solve  the  following  triangles,  the  first  two  elements  being  given 


33.  c  =  0.02934,  A  =  31°  14'.2. 

34.  c  =  4.6136,     B  =  47°  15'.6. 

35.  c  =  436.53,     A  =  74°  10'.6. 
/   36.  0  =  0.96724,  B  =  40°  40'.2. 

37.  0  =  110.97,      a  =  67.291. 

38.  0  =  1843.7,      &  =  618.42. 

39.  c  =  8226.5:      a  =  814.33. 

40.  0  =  0.03672,    6=0.01296. 
./  41.  0  =  4.8293,      b  =  0.31435. 


^  =  58°45'.8;    a  =  0.015215;  &  =  0.025086. 

^  =  42°44'.4;  a  =  3.1311  ;  6  =  3.3885. 

B=15°49'Ai  a  =  419.98;  6  =  119.03. 

A  =  49°  19'.8  ;  a  =  0.73363  ;  b  =  0.63036. 

A  =  37°  19'.8  ;  ^  =  52°  40'.2  ;  b  =  88.236. 

A  =  70°  24M  ;  5  =  19°  35'.9  ;  a  =  1736.9. 

^  =  81°50'.5;  .5=    8°    9'.5 ;  6  =  116.74. 

A  =  69°  19'.9  ;  B  =  20°  40M  ;  a  =  0.034357. 

A  =  86°  16'.1 ;  ^  =    3°  43'.9 ;  a  =  4.8191. 


RKJHT   PLANE   TRIANGLES. 


29 


42.  a  =  43.148,  6  =  84.107. 

43.  a  =  769.28,  ft  =  61.86. 

44.  a  =  7642.5,  ft  =864.7. 

45.  a  =  0.04326,  ft  =  0.54318. 

46.  a  =  903.64,  A  =  22°  lO'.S. 

47.  ft  =0.47922,  A  =  62°  16' A. 

48.  a  =  8.4642,  i?  =  30°  16'.4. 

49.  ft  =  18.430,  B  =  65°  15'.6. 


.^  =  27°    9'.5;  i?  =  62°50'.5;  c  =  94.630. 

.  ^  =  86°    6'.6;  B=    3°54'.4;  c  =  761.06. 

.  ^  =  83°32'.7;  B=    6°27'.3;  c  =  7691.3. 

.  A=   4°33'.2;  I?  =  85°  26'.8  ;  c  =  054489. 

•.  5  =  67°49'.7;  ft  =  2217.4  ;  c  =  2394.6. 

•.  i?  =  27°43'.6;  a  =  0.91176  ;  c  =  1.0300. 

:  A  =  59°  43'.6  ;  ft  =  4.9409  ;  c  =  9.80075. 

.   A  =2A°  44'.4  ;  a  =  8.4954  ;  c  =  20.299. 


Solve  the  isosceles  triangles  (Fig.  16)  in  the  following  examples,  the  first 
two  elements  being  given  : 


50.  a =57.906,  ft =62.736. 

51.  a=3.4782,  a=20°20'.6. 

52.  rt  =99.674,  /3=40°30'.4. 

53.  6=0.96042,  a  =  70°10'.4. 

54.  ft  =  1146.48,  i3=80°36'.4. 

55.  a=87.904,  j9=46.812. 

56.  6=6.9044,  p= 5.7806. 

57.  J9  =  18.478,  a=  37°  19'.8. 

58.  i)=0.46424,  /3=100°36'.8. 


.-.  a=  67°12'.05;  /8=66°  35'.9  ;  p=48.673. 
.-.  ^  =  139°18'.8;  6=6.5224;  ;)  =  1.209L 
.-.  a=  69°44'.8;  6  =  69.008; 
.-.  /3=  39°39'.2;     a  =  1.4158; 

a  =  886.24; 

i3=115°38'.8 


.-.   a=  49°41'.8 

.-.   a=  32°  10 '.6 

.-.   a=  59°   9'.2;  ^3=  61°41'.6 

.-.  ^  =  105°20'.4;  a=30.471; 

.-.  a=  39°41'.6;  a  =  0.72690; 


p  =  93.610. 
p=  1.3319. 

jr)=675.87. 
6  =  148.806. 
a =6.7330. 
6=48.458. 
6  =  1.11865. 


4 


CHAPTER   III. 

TRIGONOMETRIC   FUNCTIONS  OF  ANY  ANGLE. 

30.    Generation  of   Angles.  —  An  angle  may  be  considered  as 
generated  by  a  line  revolving  about  a  fixed  point,  the  vertex ; 

thus  OA  revolving  about  0  in  the 
direction  a,  to  the  position  OB^  de- 
scribes the  angle  A  OB.  The  side  of 
the  angle /rom  which  the  revolution 
takes  place  is  called  the  initial  side, 
and  that  to  ivhich  the  describing  line 
moves  is  called  the  terminal  side. 
The  letters  describing  the  initial  side  are  Avritten  first  in  the 
symbol  of  the  angle,  so  that  the  angle  A  OB  is  one  in  which 
the  motion  is  from  OA  to  OB. 


J 


Fig.  24. 


31.  Direction  of  Measurement.  —  The  revolving  line  can 
move  from  OA  to  OB  either  in  the  direction  marked  a  or  in 
that  marked  h.  The  former  motion,  contrary  to  that  of  the 
hands  of  a  watch,  is  arbitrarily  considered  positive  and  the 
latter  negative.  Thus  if  the  angle  a;,  between  OA  and  OB,  is 
30°,  the  angle  AOB  is  either  +  30°  or  -  330°. 

Any  angle  has  two  measures  less  than  360°,  one  positive  and 
the  other  negative,  their  numerical  sum 
90°  being  360°. 

32.  Quadrants.  —  For  convenience 
the  measuring  circle  is  divided  into 
four  parts  called  quadrants.,  as  in  the 
figure.  An  angle  is  in  the  first  quad- 
rant when  its  value  lies  between  0° 
and  90°;  in  the  second,  between  90° 
and  180°;  in  the  third,  between  180° 
30 


TRIGONOMETRIC  FUNCTIONS  OF  ANY  ANGLE.  81 

and  270°;    in  the  fourth,  between   270°  and  360°.      Angles 
between  0°  and   —  90°  are  in  the  fourth  quadrant ;    between 

-  90°  and  -  180°,  in  the  third ;  between  -  180°  and  -  270°, 
in  the  second ;  between  —  270°  and  —  360°,  in  the  first. 

Also,  an  angle  between  zero  and  ^tt  is  in  the  first  quad- 
rant ;  between  J  ir  and  tt,  in  the  second ;  between  ir  and  |  tt, 
in  the  third ;  and  between  |  tt  and  2  tt,  in  the  fourth. 

33.  Complement  and  Supplement.  —  Two  angles  are  said  to 
be  complementary  Avhen  their  algebraic  sum  is  90°,  as  60°  and 
30°,  120°  and  -30°,  260°  and  -170°;  and  supplementary 
when  their  algebraic  sum  is  180°,  as  120°  and  60°,  230°  and 

-  50°,  300°  and  -  120°. 

Note.  — In  Fig.  2,  ^  is  the  sine  of  B ;  that  is,  it  is  the  sine  of  the  comple- 
h 

ment  of  A,  and  hence  it  is  called  the  cosine  of  A. 

Since  -|-7r  corresponds  to  90°,  and  tt  to  180°,  two  angles  are 
complementary  when  the  algebraic  sum  of  their  circular  meas- 
ures is  ^  TT,.  and  supplementary  when  it  is  tt. 

1.  The  complement  of  200°  is  90°  -  200°  =  -  110°. 

2.  The  complement  of  90°  +  x  is  90°  -  (90°  +  x)  =  -x. 

3.  The  supplement  of  2u0°  is  180°  -  200°  =  -  20°. 

4.  The  supplement  of  270°  +  x  is  180°  -  (270°  +  x)  =  -  90°  -  x. 

5.  The  complement  of  -^^  tt  is  i  tt  —  y^^  tt  =  —  |  tt. 

6.  The  supplement  of  |  tt  is  tt  —  |  tt  =  —  |7r. 

Show  that  the  complement  of  the  first  angle  of  each  of  the  following  pairs  is 
equal  to  the  second  angle  : 

7.  145°  and  -55°  ;  300°  and  -210° ;  -70°  and  +160°  ;  -200°  and  +290°. 

8.  180°  -  X    and     -  90°  +  x  ;    270°  -  x    and    -  180°  +  x ;    360°  -  x   and 

-  270°  +  X. 

9.  \  IT  and  \ir ;   f  tt  and  —  tt  ;  rr  —  x  and  x  —  \ir  ',  lir  +  x  and  —  ^  tt  —  x. 

Show  that  the  supplement  of  the  first  angle  of  each  of  the  following  pairs  is 
equal  to  the  second  angle  : 

10.  145°  and  35°  ;  225°  and  -  45°  ;  -  160°  and  340°  ;  -  70°  and  250°. 

11.  270°  -  X  and  -  90°  +  x  ;  90°  +  x  and  90°  -  x  ;  x  -  90°  and  270°  -  x. 

12.  \  IT  and  |  tt  ;  §  tt  and  -  |  tt  ;  x  —  tt  and  27r— x;  |7r  +  x  and  —  \-ir  —  x. 

134.  General  Measure  of  an  Angle.  — The  line  OA  may  be 
brought  into  the  position  OB  by  revolving  either  through  the 


32  PLANE   AXD   ANALYTICAL   TRIGONOMETRY. 

number   of    complete   revolutions   in   either    direction.      The 

general  measure  of  the  angle  A  OB 
is  then  not  x,  but  x+nZGO°^  where 
n  is  any  whole  number,  positive  or 
negative. 

The  general  circular  measure  of 
""'- — '^  the    angle    whose    circular    measure 

^'°'  ^^*  less  than  2  tt  is  a:  would  be  a;  +  2  titt, 

since  2  tt  corresponds  to  a  complete  revolution. 

1.  Show  that  1000°  is  in  the  fourth  quadrant.* 

1000°  =  720°  +  280°  =  2  X  360°  +  280°,  two  complete  revolutions  and  280° 
beyond  ;  280°  lies  in  the  fourth  quadrant. 

2.  Show  that  —  3000°  is  in  the  third  quadrant. 

-  3000°  =  -  2880°  -  120°  =  8(-  360°)  -  120°,  eight  complete  revolutions 
and  120°  beyond  in  the  negative  direction  ;  —  120°  lies  in  the  third  quadrant. 

3.  Show  that  -  (8  w  +  |)  is  in  the  first  quadrant. 

A 

-(8n  +  f)=2nx27r  +  j'^7r,  2n  complete  revolutions  and  -^^ ir  beyond ; 
2 
-j3jj  TT  is  in  the  first  quadrant. 

4.  Show  that  1500°  is  in  the  first  quadrant,  2690°  in  the  second,  2720°  in 
the  third,  2100°  in  the  fourth. 

5.  Show  that  —  010°  is  in  the  second  quadrant,    -  1100°  in  the  fourth, 
-  1400°  in  the  first,  -  1920°  in  the  third. 

6.  Show  that  -(10n4-6^  is   in  the  third  quadrant,  -(12n  +  2^  in  the 
IT  ^  3 

second,  t  (8  w  4-  7)  in  the  fourth,  |  tt  (3  n  +  2)  in  the  third, 

7.  Show  that  f  tt  (10  n  —  i)  is  in  the  fourth  quadrant,  f  tt  (15  n  —  f )  in  the 
third,  f  TT  (  -  9  n  -  f  )  in  the  third,  ^  tt  (10  w  —  9)  in  the  first. 

8.  Show  that  ^  (9  n  + 1")  will  lie  in  the  third  or  in  the  first  quadrant, 

o 
according  as  n  is  odd  or  even. 

9.  Show  that  the  general  circular  measure  of  0°  is  2  nir,  and  not  wr. 

10.  Show  that  the  general  circular  measure  of  90°  is  (2  w  +  ^)  ir ;  of  180°, 
(2n  +  l)ir;  of  270°,  (2  n  +  |)7r. 

11.  If  a;  =  60°,  show  that  one  third  of  the  general  measure  of  x  will  be  20°, 
140°,  and  260°,  the  terminal  side  of  the  angle  for  all  values  of  i  x  greater  than 
260°  falling  in  one  of  these  positions. 

We  have,  using  the  general  measure,  «  +  w  360°, 

a:  =  60°,  420°,  780°,  1140°,  1500°,  1860°,-. 
.-.  ia;  =  20°,  140°,  260°,     380°,     500°,     620°,..- 
or  I  a;  =  20°,  140°,  260°,       20°,     140°,     260°,... 

if  we  reduce  the  values  oi  ^x  that  are  greater  than  360°  to  others  less  than  360° 
by  subtracting  some  multiple  of  360°. 

*  That  is,  show  that  when  the  angle  is  1000°  the  terminal  side  will  lie  in  the 
fourth  quadrant. 


TRIGONOMETRIC  FUNCTIONS  OF  ANY  ANGLE. 


33 


12.  Jf  X  =  45°,  show  that  ]  x  will  be  15",  136°,  255°,  three  values. 

13.  If  X  =  20°,  show  that  I  x  will  be  5°,  95°,  185°,  275°,  four  values. 

14.  If  X  =  60°,  show  that  I  x  will  be  10°,  70°,   130°,  190°,  250°,  310°,  six 


values. 


1  r>i° 

15.   If  X  =  m°,  show  that  -x  will  have  n  values  less  than  3G0°,  as  — 

n  n 


m°      360'^ 
n  n 


wt°      720° 
n         n 


rn^      (n- 1)360° 
n  n 


35.  The  definitions  of  the  trigonometric  ratios  in  Art.  8  are 
applicable  only  to  angles  less  than  90°.  We  shall  now  con- 
sider the  more  general  definitions,  of  which  those  in  Art.  8 
are  special  cases. 

36.  Map  Drawing  by  Coordinates.*  —  Let  ABCD  be  a  field 
whose  map  is  wanted.  From  any  point  0  in  the  field,  measure 
the  distances  Oa,  Oh^  Oc,  and  Od^  and  also  measure  the  dis- 
tances aA,  bB,  cO,  and  dD,  at  right  angles  to  X'OX,  Lay  off 
on   the   paper   a   line   X' X   of 

indefinite   length,  and  take  on  j 

it   some   point    0  to   represent  I 

the  point  0  in  the  field.  Lay 
off  Oa  according  to  some  con- 
venient scale  ;  thus  if  Oa  were  _x 
200  feet,  and  the  scale  were  20 
feet  to  1  inch,  we  would  on  the 
map  make  Oa  10  inches  long. 
Then  draw  the  line  aA  perpen- 
dicular to  OX  on  the  proper 
side  of  OX^  and  lay  off  on  it 
the  distance  corresponding  to 
aA  according  to  the  same  scale,  thus  locating  the  point  A. 
The  other  points  would  be  located  in  a  similar  manner. 

Since  Oa  and  Oc  are  measured  from  0  in  contrary  direc- 
tions, and  a  A  and  cC  are  measured  on  opposite  sides  of  X' X^ 
there  is  danger  of  laying  them  off  in  the  wrong  direction  ; 
hence  their  directions  must  be  carefully  distinguished. 

37.  Coordinates.  — The  distance  Oa,  measured  along  X'OX, 
is  called  the  abscissa  of  the  point  A  ;  aA^  measured  parallel  to 

*  This  is  called  the  method  of  offsets. 

CBOCK.   TRIG.  —  3 


Fig.  27. 


84 


PLANE  AND  ANALYTICAL   TRIGONOMETRY. 


Fig.  28. 


Y'OY^  the  ordinate  of  A;  and  the  two  distances  Oa  and  aA^ 
the  coordinates  of  A.  The  line  X'OX  is  called  the  axis  of 
abscissas ;  the  line  Y'OY,  the  axis  of  ordinates;  and  the  point 

0,  the  origin  of  coordinates. 

The  abscissa  of  a  point  is  its 
distance  from  the  axis  of  ordi- 
nates measured  on  a  line  parallel 
to  the  axis  of  abscissas. 

The  ordinate  of  a  point  is 
its  distance  from  the  axis  of 
abscissas  measured  on  a  line 
parallel  to  the  axis  of  ordinates. 
The  abscissa  is  positive  when 
the  point  is  on  the  right  of  the 
axis  of  ordinates,  and  negative 
when  it  is  on  the  left;  the  ordi- 
nate is  positive  when  the  point  is  above  the  axis  of  abscissas,  and 
negative  when  it  is  below.  If  we  consider  the  abscissas  as 
measured  from  Y'OY,  and  the  ordinates  from  X'OX,  they  will 
be  positive  when  measured  to  the  right  and  upward  respec- 
tively. 

Using  the  customary  notation  for  directed  lines,*  Oc  will 
represent  a  line  measured  from  0  to  <?,  and  cO  will  be  measured 
from  c  to  0.  The  line  cO  measured  to  the  right  is  positive, 
and  Oc  to  the  left  is  negative.  Hence  the  coordinates  of  0 
are  Oc  and  cC,  or  —  cO  and  —Co.  For  brevity,  the  coordinates 
of  a  point  are  written  in  a  parenthesis  with  a  comma  between 
them,  the  abscissa  being  written  first ;  thus  the  point  D  is 
called  the  point  (^Od,  dD). 

The  ordinate  of  any  point  on  X'OX  is  zero,  the  abscissa  of 
any  point  on  Y'OY  is  zero,  and  both  coordinates  of  the  origin 
are  zero. 

The  signs  of  the  numerical  coordinates  of  points  in  the 
different  quadrants  are  as  follows  : 


Quadrant 

Abscissa 

Ordinate 


I.     11. 

+      - 


III. 


IV. 

4- 


*  See  Art.  2. 


TRIGONOMETRIC  FUNCTIONS  OF  ANY  ANGLE. 


35 


38.  Distance  of  a  Point  from  the  Origin. — Represent  the 
abscissa  of  the  point  by  a,  its  ordinate  by  o,  and  its  distance 
from  the  origin  by  h.     Then 


h  =  Va2  +  o\ 

since  h  is  the  hypotenuse  of  a  right  triangle  whose  sides  are  a 
and  0.  Although  h  may  be  either  positive  or  negative,  it  will 
be  sufficient  for  our  purposes  to  treat  it  as  being  always 
positive. 


Fig.  29. 


39.  Trigonometric  Ratios.  —  Take  the  origin  of  coordinates 
at  the  vertex  of  the  angle  and  the  initial  side  as  the  axis  of 
abscissas.  From  any  point  B  on  the  terminal  side  of  the  angle, 
draw  AB  perpendicular  to  the  initi^  side  ;  denote  the  abscissa 
OA  of  the  point  by  a,  its  ordinate  AB  by  o,  and  its  distance  OB 
from  the  origin  by  A.  The  general  definitions  of  the  trigono- 
metric ratios  are  : 


The  sine  of  the  angle 

The  cosine  of  the  angle 
The  tangent  of  the  angle 
The  cotangent  of  the  angle 
The  secant  of  the  angle 
The  cosecant    of  the  angle 


ordinate 

o 
Yi 

a 
~  h 

o 

~  a 

a 

~  o 

h 

~  a 

h 

~  o 

distance  ~ 
abscissa 

distance  ' 
ordinate 

abscissa 
abscissa 

ordinate  ~ 
distance 

abscissa  ~ 
distance 

ordinate  ~ 

(1) 


36  PLANE  AND  ANALYTICAL   TRIGONOMETRY. 

Note.  —  The  origin  is  always  at  the  vertex  of  the  angle;  the  axis  of 
abscissas  always  coincides  with  the  initial  side ;  and  the  positive  direction  of  the 
axis  of  ordinates  is  along  the  line  that  makes  an  angle  of  +  90°  with  the  initial 
side. 

Prove  that  the  following  equations  are  true,  using  Eqs.  (1): 
1.         ^^^  ^        =  cosec  X. 


V  sec*  X  —  1 

sec  a;  =  - ; 
a 
h 


-1     J? 


\/sec2  X  -  1     ^  fi^      1      Vh^ 


-  =  cosec  X. 


a" 


2.  sec  X  cos  X  =  1.  7.  tan  x  cot  x  =  1. 

3.  cosec  X  sin  X  =  1.  8.  sin^x  +  cos^x  =  1. 

4.  cosec*  X  =  1  +  cot2  x.  9.  sec*  x  =  1  +  tan-^  x. 

6.      ,*^!i^„  =  si»r«.  10.  vlli2t!^=seca.. 
V 1  +  tan"'^  X                                                       cot  x 


6.    Vcosec-^^-l^eosx.  11.   J  l  +  cot*x    ^^^^^^ 

cosec  X  '  cosec  "-^  x  —  1 

12.  (tan  X  —  cot  x)  (tan  x  +  cot  x)  =  — ^  -^  =  sec*  x  —  cosec*  x. 

13.  (tanx  +  cotx)  sinx  cosx  =  1. 

Construct  geometrically  the  angles,  and  compute  the  corresponding  ratios  in 
the  following  examples :  * 


Quadrant.     Sin. 

Cos. 

Tan. 

Cot. 

Sec. 

Cosec. 

14. 

sinx=  +  |. 

I. 

— 

+  f 

+  |. 

+  |. 

+  1- 

+  f. 

'       II. 

— 

-1- 

-f. 

-f 

-|. 

+  f- 

15.. 

sinx==-i. 

III. 

— 

-fV2. 

+  iV2. 

+  2\/2. 

-|V2. 

-3. 

IV. 

— 

+  f\/2. 

-iV2. 

-2\/2. 

+fv^. 

-3. 

16. 

cosx=  +  |. 

I. 

+  hVs. 



+  V3. 

+  iV3. 

+  2. 

+  |V3. 

IV. 

-W3. 

— 

-V3. 

-iV3. 

+  2. 

-|>/3. 

17. 

cosx=-}. 

II. 

+  fV2. 

— 

-2V2. 

-i\^. 

-3. 

+  IV2. 

III. 

-f\/2. 

— 

+  2\/2. 

+  iV2. 

-3. 

-|V2. 

18. 

tanx=+^. 

I. 

+  iV5. 

+  f\/5. 

— 

+  2. 

+  IV5. 

+  V5. 

III. 

-^V5. 

-fV5. 

— 

+  2. 

-^>/5. 

-V5. 

19. 

tan  x=  -2. 

II. 

+  |V5. 

-|\/5. 



-h 

-V5. 

+  1V5. 

IV. 

-f\/5. 

+  iV5. 

— 

-h 

+  V5. 

-|Vg. 

*  See  Arts.  11  and  15.    If  sin  x  is  positive,  0  must  be  positive,  since  h  is 
always  positive,  and  the  angle  lies  in  quadrants  I.  and  II. 


TRIGONOMETRIC  FUNCTIONS  OF  ANY  ANGLE.    37 


Quadrant,     Sin. 

Cos. 

Tan. 

Cot. 

Sec. 

Cosec. 

cotx=+|. 

I. 

+  f 

fl. 

+  |. 

— 

+  f- 

+  !• 

III. 

-f. 

I. 

+  |. 

— 

-i- 

_5 
3* 

cot  2= -3. 

IT. 

+tVv1o. 

-^VTo. 

-f 

— 

-jVTo. 

+  Vio. 

IV. 

-T^.VIo. 

+^vlo. 

-h 

— 

+  ^VIo. 

->/io. 

sec  a;  =+3. 

I. 

4f>/2. 

+f 

+  2\/2. 

+  iV2. 

— 

+IV2. 

IV. 

-fV2. 

+1. 

-2V2. 

-\V2. 

— 

-|v^. 

seca:=  — ^. 

II. 

+  !• 

-h 

-|. 

-I 

— 

+1- 

III. 

-f. 

-f. 

+  t. 

+  1- 

— 

-l. 

cosecx=+-U. 

I. 

+  tV 

+  H- 

+  tV 

+v. 

+f|. 

— 

II. 

+A. 

-H- 

-A- 

-V-. 

-ii- 

— 

coseca;=— y-. 

III. 

-^T. 

-if 

+iV 

+¥. 

-If. 

— 

IV. 

-/5- 

+  lf 

-^i. 

-¥• 

+  l|. 

— 

21. 
22. 
23. 
24. 
25. 


40.  Trigonometric  Functions.  —  One  quc^ntity  is  said  to  be  a 
function  of  anotlier  when  it  depends  upon  the  latter  for  its 
value.  Thus,  if  2/  =  sin  a;,  ?/  is  a  functioii  of  x,  since  it  depends 
upon  X  for  its  value,  any  change  in  the  value  of  x  producing  a 
change  in  the  value  of  y. 

The  trigonometric  functions  are  the  sine,  cosine,  tangent, 
cotangent,  secant,  cosecant,  versed  sine,  coversed  sine,  and 
suversed  sine.     The  last  three  are  defined  by  the  eq.uations  : 


The  versed  sine  is  vers  a;  =  1  -  cos  a? 

The  coversed  sine  is      covers  a;  =  1  -  sin  a; 
The  suversed  sine  is     suvers  a;  =  1  +  cos  a;. , 


(1) 


41.  Geometrical  Representation  of  the  Functions.  —  In  Fig.  30 
let  the  radius  OB,  of  the  circle  described  about  the  vertex  0 
of  the  angle  AOB  as  a  center,  be  unity,  and  let  the  angle  AOY 
be  equal  to  90°.  NM  and  FB  are  tangent  to  the  circle  at 
X  and  F  respectively ;  the  triangles  OAB,  OXM,  and  OYB, 
are  right-angled ;  and  the  angle  YB  0  is  equal  to  the  given 
angle  AOB.  Then  the  trigonometric  functions  of  the  angle 
AOB  are  represented  by  the  lines  shown  in  the  figure.  For, 
in  Figs.  2  and  29,  B  is  any  point  on  the  terminal  side  OB 
of  the  angle  AOB,  and  therefore  we  may  choose  the  position 
of  B  so  that  OB,  or  A,  shall  be  equal  to  unity.  Comparing 
Fig.  30  with  Figs.  2  and  29,  and  using  the  definitions  in 
Arts.  8,  39,  and  40,  we  see  that 


38 


PLANE   AND   ANALYTICAL   TRIGONOMETRY. 


sin  AOB 
cos  AOB 
tan  AOB 
cot  AOB 
sec  AOB 

cosec  AOB 

vers  AOB 
covers  AOB 
suvers  AOB 


0      AB 
h      OB 

•  AB, 

a      OA 
h      OB 

-.OA, 

0      AB 
a      OA 

XM 
OX 

=  XM, 

a      OA 

CB 

YD 

0      AB 

00 

or 

h      OB 
a      OA 

OM 
OX 

--0M, 

h      OB 

OB 

OB 

YD. 


0  AB^  00  ^  OY  ^  ^^' 

1  -  cos  AOB  =  OX  -0A  =  AX. 
1  -  sin  AOB  =  OY  -00=  CY. 
1  +  cos  AOB  =  XO  +  0A  =  X'A. 


The  trigonometric  functions  are  ratios^ — pure  numbers, — 
and  are  represented  by  these  lines  in  the  circle  whose  radius  is 

unity  ;  that  is,  they  are  actu- 
ally equal  to  the  ratios  of  these 
lines  to  the  radius. 

If,  with  a  radius  of  unity 
and  the  vertex  of  the  angle  as 
the  center,  a  circle  be  described 
and  two  tangents  be  drawn,  one 
Avhere  the  initial  side  OA  cuts 
the  circle,  and  the  other  at  a 
distance  of  +  90°  from  this 
point  (at  X  and  Y  respec- 
tively), the  trigonometric  functions  will  be  represented  as 
follows : 

The  sine  of  an  angle  will  he  the  perpendicular  distance  from 
the  point  where  the  terminal  side  of  thi»  angle  cuts  the  circle,  to 
the  initial  side,  produced  if  necessary  ;  positive  when  it  is  above, 
and  negative  when  below,  the  initial  side.  Thus  sin  A  OB  =  AB, 
sin  AOJI  =  GH,  sin  A  OK  =  aK,  sin  A  OL  =  AL.  AB  and 
6rl£,  above  X'  OX,  are  positive,  while  GK  and  AL  are  nega- 
tive, being  below  X'  OX.     The  sine  is  therefore  positive  when 


TRIGONOMETRIC   FUNCTIONS   OF   ANY   ANGLE.  39 

the  angle  is  in  the  first  or  second  quadrant,  and  negative  when 
it  is  in  the  third  or  fourth. 

The  cosine  will  be  the  distance  from  the  center  to  the  foot  *  of 
the  sine  ;  positive  when  measured  to  the  rights  and  negative  to  the 
left,  of  the  center.  Thus  cosAOB:=OA,  cos  AOI£=  OG, 
cos  A  OK  =  0G-,  cos  A  0L=  OA.  OA,  measured  to  the  right 
of  the  center,  is  positive,  while  0(r,  measured  to  the  left,  is 
negative.  The  cosine  is  therefore  positive  when  the  angle  is 
in  the  first  or  fourth  quadrant,  and  negative  when  it  is  in  the 
second  or  third. 

The  tajigent  will  be  the  distance  along  the  line  tangent  to  the 
circle  at  the  point  where  the  initial  side  cuts  the  circle,  from  this 
point  to  the  point  where  this  tangent  is  cut  by  the  terminal  side  of 
the  angle,  produced  if  necessary ;  positive  when  measured  above, 
and  negative  when  beloiv,  the  initial  side.  Thus  tan  ^0^  =  XM, 
tan  A  011=  XJSr,  tan  A  0K=  X^,  tan  AOL  =  XK  XM,  above 
X'  OX,  is  positive,  and  XJSf,  below  X'  OX,  is  negative.  There- 
fore the  tangent  is  positive  when  the  angle  is  in  the  first  or 
third  quadrant,. and  negative  when  it  is  in  the  second  or  fourth. 
^  The  cotangent  will  be  the  distance  along  the  second  tangent 
(FYD)  from  the  point  of  tangency  to  the  point  ivhere  this  line  is 
cut  by  the  terminal  side  of  the  angle,  produced  if  necessary  ;  posi- 
tive when  measured  to  the  right,  and  negative  to  the  left,  of  the 
point  of  tangency.  Thus  cot  A  OB  =  YD,  cot  A  OH  =  YF, 
cot  AOK=  YD,  cot  AOL  =  YF.  YD,  measured  to  the  right, 
is  positive,  and  YF,  measured  to  the  left,  is  negative.  There- 
fore the  cotangent  is  positive  when  the  angle  is  in  the  first  or 
third  quadrant,  and  negative  when  it  is  in  the  second  or  fourth. 

Note,  — The  positive  directions  of  measurement  are  above  X'OX  and  to 
the  right  of  TOY,  and  the  negative  are  below  X'OX  and  to  the  left  of  Y'OY. 

The  secant  ivill  be  the  distance  from  the  center  along  the  ter- 
minal side  of  the  angle,  produced  if  necessary,  to  its  point  of 
intersection  with  the  taiigent  at  the  point  of  intersection  of  the 
initial  side  with  the  circle;  positive  when  measured  along  the 
side  itself,  and  negative  when  along  the  side  produced.  Thus 
sec  AOB=  OM,  sec  A  0H=  OX,  sec  A  0K=  OM,  sec  AOL=  ON. 

*  The  foot  of  the  sine  is  the  point  where  the  perpendicular  line  representin  ; 
the  sine  cuts  the  initial  side,  produced  if  necessary.  |  % 


40  PLANE   AND   ANALYTICAL   TRIGONOMETRY. 

Since  sec  A  OB  and  sec  AOL  are  measured  along  the  terminal 
side  itself,  they  are  positive.  The  terminal  sides  (^OH  and 
OK}  of  the  angles  A  OH  and  A  OK  must  be  produced  in  order 
that  they  may  intersect  the  tangent  line  iV7l[f,  and  therefore 
^ec  A  Off  and  sec  A  OK  are  negative.  Hence  the  secant  is 
positive  when  the  angle  is  in  the  first  or  fourth  quadrant,  and 
nes^ative  when  it  is  in  the  second  or  third. 

The  cosecant  will  he  the  distance  from  the  center  along  the 
terminal  side^  produced  if  necessary^  to  its  intersection  with 
the  second  tangent^  FYD ;  positive  when  measured  along  the 
side  itself^  and  negative  when  along  the  side  produced.  Thus 
GosQcAOB=  OB,  cosec  ^Oir=  OF,  cosec  A  OK  =  OB, 
cosec  A  OL  =  OF,  Since  cosec  A  OB  and  cosec  A  OH  are  meas- 
ured along  the  terminal  side  itself,  they  are  positive,  while 
cosQc AOK  and  cosqgAOL,  measured  along  the  side  pro- 
duced, are  negative.  Therefore  the  cosecant  is  positive  when 
the  angle  is  in  the  first  or  second  quadrant,  and  negative  when 
it  is  in  the  third  or  fourth. 

The  versed  sine  (1  —  cos  x)  will  he  the  distance  from  the  foot 
of  the  sine  to  the  point  where  the  initial  side  cuts  the  circle; 
always  positive,  hecause  cosx  can  never  he  greater  than  the' 
radius,  or  unity.  Thus  vers  A  OB  =  AX,  vers  A  Off  =  GX, 
Yeis AOK=  ax,  Yers AOL  =  AX. 

The  coversed  sine  (1  —  sin  x)  ivill  he  the  distance  from  the 
point  C  or  P,  where  a  line  drawn  through  the  point  of  intersection 
of  the  terminal  side  and  the  circle  parallel  to  the  initial  side  cuts 
Y'OY,  to  the  point  Y;  always  positive,  since  sinx  can  never  he 
greater  than  the  radius,  or  unity.  Thus  covers  A  OB  =  CY, 
covers  AOff=  CY,  covers  AOK=  PY,  covers  AOL  =  FY. 

The  suversed  sine  (1  +  oos  x)  will  he  the  distance  from  the 
point  yj ,  where  the  initial  side  produced  cuts  the  circle,  to  the 
foot  of  the  sine ;  always  positive,  since  cos  x  can  never  he  alge- 
hraically  less  than  minus  unity.  Thus  suvers  AOB  =  X^A, 
suvers  A Off=  Xa,  suvers  A 0K=  X' a,  suvers  AOL  =  X'A. 

Note. — These  lines  represent  th^  trigonometric  functions,  only  when  the 
radius  of  the  circle  is  unity.  If  the  radius  differs  from  unity,  the  functions  are 
equal  to  the  lengths  of  these  lines  divided  by  the  radius. 

42.    Changes  in  the  Values  of  the  Functions.  —  Let  OX  be 

the  initial  side   of  the   angle,  and  let  the   terminal  side   first 


TRIGONOMETRIC   FUNCTIONS  OF   ANY  ANGLE. 


41 


r 

Y              cot 

D 

\ 

^ 

.    1 

C              E\ 

A 

f 

/ 

/      cos 

^ 

1 

X 

\ 

0          y^ 

OV            ^ 

ven 

1 

^ 

■-^ 

P              1/ 

H 

coincide  with  OX^  and  then,  in  revolving  about  0,  come  into 
the  positions  OM,  OY,  OH,  OX',  OK,  (9^^  ON,  and  OX,  and 

let  us  consider  the  resulting  changes  in  the  values  of  the  sine 
and  of  the  tangent. 

The  sine  of  0°,  the  terminal  side  coinciding  with  OX,  is 
zero.  As  the  angle  increases, 
the  sine,  being  positive,  also  in- 
creases (sin  A  OB  =  AB},  until 
at  90°  it  is  equal  to  the  radius, 
or  +l(sinJL07=  OF).  The 
sine  then  decreases  {ain  A  Oil 
=  G-H^,  still  being  positive ; 
and  at  180°  it  is  zero,  the  ter- 
minal side  coinciding  with  OX' . 
The  sine  then  becomes  negative, 
and  decreases  algebraically,  in- 
creasing numerically  (p\nAOK=  (tK),  until  at  270°  it  is 
equal  to  the  radius,  or  —  1  (sin  ^ OF'  =  OF').  It  then  in- 
creases algebraically,  decreasing  numerically  (sin^OX  =  ^X); 
and  at  360°  it  again  becomes  zero. 

The  tangent  of  0°  is  zero  ;  the  tangent  then  becomes  posi- 
tive, and  at  90°  it  is  infinite,  the  terminal  side  being  parallel 
to  XM;  then  negative,  and  at  180°  it  is  zero  ;  then  positive, 
and  at  270°  it  is  infinite  ;  then  negative,  and  at  360°  it  is  zero. 
Just  before  the  terminal  side  reaches  the  position  0  F,  the 
tangent  is  positive,  and  just  after,  it  is  negative  ;  therefore 
the  tangent  of  90°  is  ±qo,  the  upper  sign  being  that  of  the 
function  of  an  angle  a  little  less  than  90°. 

The  table  gives  the  values  of  the  functions  of  0°,  90°, 
180°,  270°,  and  360°,  and  their  signs  in  quadrants  I.,  II.,  III., 
and  IV. : 


Fig.  31. 


0°. 

I. 

90°. 

II. 

lS()o. 

III. 

270°. 

IV. 

360°. 

sin. 

0 

+ 

+  1 

p 
+ 

0 

_ 

-1 

_ 

0 

COS. 

+  1 

+ 

9- 

— 

-1., 

— 

0 

+ 

+  1 

tan. 
cot. 

0 

CO 

+ 

CO 

0 

_ 

0 

CO 

+ 
+ 

CO 

0 

— 

0 

CO 

sec. 

+  1 

+ 

CO.. 

-  1 

— 

00 

+ 

+  1 

cosec. 

00 

+ 

+  1 

+ 

QO 

- 

- 1 

- 

QO 

^' 


v^V 


42  PLANE   AND   ANALYTICAL   TRIGONOMETRY. 

43.  Limiting  Values  of  the  Functions.  —  The  sine  and  cosine 
may  have  any  value  between  +1  and  —1,  but  they  cannot  have 
a  value  numerically  greater  than  unity. 

Tlie  tangent  and  cotangent  may  have  any  value  between 
+  c»  and  —00  ;  that  is,  no  matter  what  a  number  may  be,  there 
will  always  be  some  angle  that  will  have  that  number  as 
the  value  of  its  tangent,  and  another  having  it  as  its  co- 
tangent. 

The  secant  and  cosecant  may  have  any  value  between  -|-1 
and  +  X),  or  —  1  and  —  oo ;  but  they  cannot  have  a  value  numeri- 
cally less  than  unity. 

The  versed  sine,  coversed  sine,  and  suversed  sine  may  have 
any  value  between  zero  and  -f  2. 

Note.  —  In  the  first  quadrant,  all  the  functions  are  positive,  and  the  sine, 
tangent,  and  secant  increase  as  the  angle  increases ;  while  the  cosine,  cotangent, 
and  cosecant  decrease  as  the  angle  increases. 

Note.  — The  functions  change  signs  only  when  they  pass  through  the  values 
zero  and  infinity. 

44.  Graphical  Representation  of  the  Functions.  —  Let  the  dis- 
tance OL  represent  360°,  so  that  1°  is  represented  hj  -^\-qOL. 
At  (7,  such  that  OQ  =^0L^  draw  a  line  perpendicular  to  OL^  and 


D  E  \F  226'  270° 


0°        SO"  46"  Wf  136"  160°  IW 


Fig.  82. 


lay  off  on  it  any  convenient  distance  0<?,  to  represent  the  sine 
of  90°,  above  the  line  OL^  since  sin  90°=  +1.  At  J.,  such  that 
OA  =  -^^OL,  lay  off  Aa  —  ^Oc,  since  sin  30°=  +  J  ;  ^t  B,  such 
that  0B  =  \  OL.  lay  off  Bh  =  (7c  VJ,  since  sin  45°  =  +  a^J  ;  at  H, 
such  that  Off=^OL,  lay  off  JIh  =  OcV^,  below  OX,  since 
sin  225°  =  —  VJ ;  and  so  on,  locating  as  many  points  a,  6,  c,  A, 
etc.,  as  may  be  necessary.  Draw  a  smooth  curve  through  0,  a, 
ft,  c,  (?,  e,  F,  A,  I,  y,  L,  and  we  have  the  sinusoid,  in  which  the 


TlUGONOxMETRlC   FUNCTIONS  OF   ANY  ANGLE. 


43 


abscissas  correspond  to  the  angles,  and  the  ordinates  to  their 

sines. 


We  might  have  taken  OL 
equal  to  the  circumference  of 
the  circle  whose  radius  is  unity, 
and  Ce  equal  to  this  radius. 
The  scale  would  then  have  been 
the  same  for  both  the  ordinates 
and  the  abscissas. 

The  graphical  representa- 
tions of  the  other  functions 
may  be  constructed  in  a  similar 
manner. 


cot 


o^ 

c 

-^ 

/ 

f 

( 

\ 

\ 

sin    \ 

^ 

1 

\ 

/ 

cos 

] 

X 

\ 

G         y 

V 

A 

p 

y\ 

^^ 

_^ 

N- 

FiQ.  38. 


45.  Two  Angles  correspond  to  Any  Given  Function.  —  In 
Fig.  33  let  the  arcs  YB,  TIT,  Y'K,  and  Y'Lhe  equal;  there- 
fore the  arcs  XB,  X'H^  X' K^  and  XL  are  equal.     Hence 

AB  =  aR=OC;   AL  =  aK=OF;    OM=ON;    OD  =  OF. 

00  is  not  equal  to  OP  since  they  have  contrary  signs,  00 
being  positive  and  OP  negative  on  account  of  their  directions. 

AB  =  ^mXOB',    aff=smXOH:; 

.-.  sin  XO^  =  sin  XOir. 
ax  =  sin  XOK ;    AL  =  sin  XOL  ; 

.-.  sin  XOK  =  sin  XOL. 

Therefore  two  angles  that  differ  by  equal  amounts  from  90°,  or 
from  270°,  will  have  the  same  sine  ;  thus  sin  (90°  +  2°)  = 
sin  (90°  -  2°),  aCT~smt270°TF7^in(270°  -  3°). 

Note.  — The  two  angles  corresponding  to  a  given  function  may  be  identical ; 
thus,  if  sin  X  =+  1,  the  only  value  of  x  is  ^0°,  or  90°  -  0°  and  90°  +  0°. 

Again  OA  =  cos  XOB  =  co^  XOL  ; 


and 


0(7  =  cos  XOH  =  cos  XOK. 


Therefore  two  angles  differing  by  equal  amounts  from  0°,  from 
180°,  or  from  360°,  will  have  the  same  cosine  ;  thus  cos  (  —  5°)  = 
cos  5°,  cos  (180° -f  5°)  =  cos  (180° -5°),  and  cos  (360° -10°)  = 
cos  10°. 

Also  XM  =  tan  XOB  =  tan  XOK; 

and  XN  =  tan  XOH  =  tan  XOL, 


44  PLANE   AND   ANALYTICAL   TRIGONOMETRY. 

Therefore  two  angles  differing  from  each  other  by  180°  will 
have  the  same  tangent ;  thus  tan  140°  =  tan  320°. 

Again  YD  =  cot  XOB  =  cot  XOK ; 

and  YF  =  cot  XOff  =  cot  XOL. 

Therefore  two  angles  differing  from  each  other  by  180°  will 
have  the  same  cotangent ;  thus  cot  200°  =  cot  20°. 

Also        +0M=  sec  XOB;    +  ON  =  sec  XOL  ; 

.-.  sec  XOB  =  sec  XOL. 
-0M=  sec  XOK ;    -  ON  =  sec  XOH ; 

.-.  sec  XOK  =  sec  XOH. 

Therefore  two  angles  differing  by  equal  amounts  from  0°,  from 
180°,  or  from  360°,  will  have  the  same  secant ;  thus  sec  (—  5°)  = 
sec  5°,  sec  (180°  -  3°)  =  sec  (180°  +  3°),  and  sec  (360°  -  5°)  = 
sec  5°. 

Again     -h  OD  =  cosec  XOB  ;    -{- OF  =  cosec  X OK ; 

.  \   cosec  XOB  =  cosec  XOK. 
-0D=  cosec  XOK;    -  OF  =  cosec  XOL  ; 

.  • .  cosec  XOK  =  cosec  XOL. 

Therefore  two  angles  differing  by  equal  amounts  from  90°,  or 

from  270°,  will  have  the  same  cosecant ;  thus  cosec  (90°  +  10°)  = 

cosec  (90°  -  10°),  and  cosec  (270°  -  60°)-  cosec  (270°  +  60°). 

The  four  angles  XOB,  XOH,  XOK,  and  XOL,  have  the  same  functions 
numerically.     Thus  if  sin  x  =  ±  |,  x  will  be  30°,  150^,  210°,  and  330°  ;  the  tirst 


EXAMPLES. 

1.  What  angle  has  the  same  sine  as  140°? 

2.  What  angle  has  the  same  sine  as  220°  ? 

3.  What  angle  has  the  same  cosine  as  330°? 

4.  What  angle  has  the  same  cosine  as  220°  ? 

5.  What  angle  has  the  same  tangent  as  230°  ? 

6.  What  angle  has  the  same  tangent  as  300°  ? 

7.  What  angle  has  the  same  cotangent  as  240°  ? 

8.  What  angle  has  the  same  cotangent  as  110°  ? 

9.  What  angle  has  the  same  secant  as  315°  ? 

10.  What  angle  has  the  same  secant  as  160°  ? 

11.  What  angle  has  the  same  cosecant  as  110°? 

12.  What  angle  has  the  same  cosecant  as  300°  ? 


Ans. 

40°. 

Ans. 

320°. 

Ans. 

30°. 

Ajis. 

140°. 

Ans. 

50°. 

Ans. 

120°. 

Ans. 

60°. 

Ans. 

290°. 

Ans. 

45°. 

Ans. 

200°. 

Ans. 

70°. 

Ans. 

240°. 

6i 


TRIGONOMETRIC   FUNCTIONS  OF   ANY  ANGLK.  45 

Find  the  values  of  0  less  than  360°  in  Exs.  (13-24)  :  ♦ 

13.  sin  5  =  - sin  200°.     Ans.     20°.            19.       cot  ^  =- cot  106°.  An8.  76°. 

14.  sin  ^  =- sin  100°.     ^ns.  260°.            20.       cot  «  =  - cot  205°.  ylns.  166°. 
16.   cos(?  =  -cosl50°.      Ans.     30°.            21.       sec ^  =  -  secl40°.  Ans.  40°. 

16.  cos<?=-cos300°.     Ans.  120°.  22.       sec  ^  =  - sec  326°.       Ans.  146°. 

17.  tan ^= -tan 350°.     A71S.     10°.  23.   cosec^  =- cosecl20°.    ^ns.  240°. 

18.  tan  0= -tan  230°.     Ans.  130°.  24.    cosec  ^  =  -  cosec  355°.  Ans.      6°. 

25.  cos  3  ^  =  +  i  y/S.     Find  three  values  of  0  less  than  180°. 

3  0  may  he  30°,  or  330°,  or  these  values  plus  any  number  of  circumferences; 
.-.  3^  =  30°,  390°,  760°,  ••.,  330°,  690°,  1050°,  -. 
.-.      0  =  10°,  130°,  250°,  ...,  110°,  230°,    360°,  ... 

Ans.  0  =  10°,  110°,  130°. 

26.  sin  2  ^  =  -  h    Find  four  values  of  0  less  than  360°. 

Ans.  106°,  285°,  165°,  345°. 

27.  tan  3  ^  =  —  1.    Find  six  values  of  0  less  than  360°. 

Ans.  45°,  165°,  285°,  105°,  226°,  345°. 

28.  sec  6  ^  =  -2.     Find  five  values  of  0  less  than  180°. 

Ans.  24°,  96°,  168°,  48°,  120^ 

29.  cot  5  ^  =  + 1.     Find  five  values  of  0  less  than  180°. 

Ans.  9°,  81",  153°,  45°,  117°. 

30.  cos4  0  =  —  i.     Find  four  values  of  0  less  than  180°. 

Ans.  30°,  120°,  60°,  150°. 

31.  sin  0  =  h.     Show  that  the  general  measure  of  0  is  (2  n  +  ^)ir  ±  i  tt. 
0  =  30°  and"  150°,  or  90°  -  60°  and   90°  +  60°,  or  90°  db  60°,  or  '^  tt  ±  A  tt. 

But  the  general  measures  of  0  are  these  values  increased  by  any  number  (;i)  of 
circumferences.     .  •.  0  =  '^  nir  -{-  ^  ir  ±.^  ir  =  (2n  +  ^)  tt  ±  i  tt. 

32.  sin  0  =  +  hV2,  tan  0  =  —  1  ;  the  general  measure  of  ^  is  2  wtt  +  l-v. 
Note  that  0  is  in  the  second  quadrant,  since  its  sine  is  positive  and  its  tan- 
gent is  negative. 

Q 

33.  cos  ^  =  —  ^    cosec  0  =  -\ ::;  the  general  measure  of  0  is  2  nrr  +  6', 

2V2 
where  6'  is  the  value  of  0  that  lies  between  ^  tt  and  w. 

34.  cos  (9  =  —  I ;  the  general  measure  of  ^  is  (2  w  +  1)t  ±  ^  tt. 

35.  sin  2  ^  =  +  i  ;    the  general  measures  of  0  are   (2  w  +  |)  tt  ±  |  tt,  and 
(2  n  +  f )  ^  ±  i  TT. 

36.  cos  3  ^  =  —  i  ;     the    general    measures    of    0    are    (2  n  +  |)  tt  ±  ^  tt, 
(2  n  +  1)  TT  ±  i  TT,  and  (2  n  +  f )  tt  i  ^-  ir. 

Construct  geometrically  (Art.  11)  the  two  angles  when 

37.  sinx=+i.                41.   tanx  =  +  2.                45.  sec  a;  =  +3. 

38.  sin  a;  =  —  ^.                 42.    tan  x=—\.                46.  sec  x=—\. 

39.  cos  a;  =  +  |.                 43.    cot  sc  =  +  |.                 47.  cosec  re  =  +  6. 

40.  cos  a;  =  —  |.                44.    cot  r  =  —  f .                48.  cosec  a:  =  —  |. 

♦  Only  one  of  the  two  answers  is  given. 


CHAPTER  IV. 

RELATIONS  BETWEEN  THE  FUNCTIONS  OF  ONE  ANGLE. 

46.    Relations  between  the  Functions  of  One  Angle. 

o2  4.^2  =  ;^2. 


or. 


A2^A2 


sin'^  dc  +  cos^  oc  —  \^ 


0      h      sma? 

tan  X  =  -  =  -  = ; 

a      a      cos  a; 

I 

sino? 


tan  05 


cos  05' 


cot  05=-  =  7 — -; 

0     tan  05 


cot  05 


COS  05 


\ 


sin  05 

A2  =  a2  +  o2; 

.  *.  sec^  05  =  1  +  tan'^  x, 

7,2  =  02  +  ^2; 
.  •.   COSeC^  05  =  1  +  COt^  05. 

h       1 
sec  05  =  - 


=1+^ 


7(2 


^^^ 


a     cos  05 

h       1 
cosec  ic  =  -  =  —. — • 
0     sinx 

Ters  a!  =  1  -  COS  x. 

corers  a;  =  1  -  sin  as. 

snyers  a;  =  1  +  cos  a;. 

46 


(1) 


(2) 

(S) 
(4) 

(5) 

(6) 
(7) 

(8) 

(9) 
(10) 

(11) 


FUNCTIONS  OF  ONE  ANGLE.  47 

Note.  — These  formulas  may  be  easily  remembered  by  the  use  of  Fig.  30, 
where 

AB^-\-  OA^  =  0B\     or      sin2  a;  +  cos2  x=l. 

.„„  ^      XM     AB      ^^       .^  ^      sin  x 

tan  X  =  — — -  =  — — ,     or       tan  x  = 

OX      OA  cos  a; 

cotar=— —  =    — ,      or       cotx  = 

OY      OC  sinx 

0.1/2  =  0X2  +  xJlf2,  or      sec2  x  =  1  +  tan2  x. 

OD'-  =  Or2  +  YD\   or  cosec2  a;  =  1  +  cot2  x. 


\  47.    To   express   One    Function   in   Terms   of    Each  of    the 

I  Others.  —  Suppose  that  we  wish  to  find  expressions  for  sin  a: 

[  that   shall   contain   only  cos  a;,  tana:,  cot  a;,  sec  a:,  and   cosec^ 

I  respectively.     From  the  preceding  article  we  have: 

^  sin^  X  +  cos^  a:  =  1,  and  cosec  x  = 


sin  a;' 


and  sin  x  = 


sin  X  =  ±  Vl  —  cos^  a^ 
1 


cosec  X 


The  other  expressions  are  derived  from  these  as  follows : 


^       =  ±  ^  from  (6). 


cosec  X  Vl  +  cot2  X 

1  ,  1  ,         tanx  ,  ,„. 

•.  sin  X  =  ±  —  =  ± —  -  =  ±  — ,  from  (3). 


V 1  +  cot2  X  L  1  Vl  +  tan2  x 

\        tan^x 


r,i^  ^      I         tanx  ,  \/sec2x  — 1    i?^^^  /c^ 

.  •.  sm  X  =  ±  •=  ± ,  from  (5). 

Vl  +  tan2  X  sec  x 

The  double  signs  are  due  to  the  fact  that  there  are  two 
angles  corresponding  to  any  given  function  ;  thus  if  cos  a;  =  |^, 
the  angle  might  be  either^  in  the  first  or  in  the  fourth  quadrant, 
and  the  sine  would  be  positive  in  the  first  case  and  negative 
in  the  second.  It  will  be  seen  that  if  any  one  of  the  functions 
'  is  given,  all  the  others  found  from  it  will  have  the  double  sign, 
except  its  reciprocaL 


48 


PLAN"£   AND   ANALYTICAL   TRIGONOMETRY. 


In  the  same  way  it  may  be  shown  that* 


1                     cot  X              1         Vcosec-^  X  —  1 

cos  X  —  v'l  —  sin^  X  — 

Vl  +  tan^  X      Vl  +  cof-^  x     sec  x           cosec  x 

• 

tan  X-        ^^^^ 

Vl-cos''x_     1     _^/sec2a.      i_             1 

Vl  -  sin2  X 

cos  X           cot  X                             Vcosec^  x  —  1 
^^^^       -     ^     -          ^    •      -Vcoscc^x      1 

sinx 

Vl  -  cos2  X     i^aii  a:      Vsec2  x  -  1 

aan  n>  —                                        — 

^     -  Vl  +  tan2  X  -  ^^  +  ^*^''  ^  -        ^^^^^  *       . 

Vl  —  sin^  X 

cosx                                    cotx            Vcosec2x-l 

1  1  Vl  4-  tan2  X       /i    .  ^^♦•2^  sec  x 

cosec  X  = =  —  —  = ■ =  V 1  -f  cot-*  X  =  —  -■ 

sinx      Vl  -cos2x  ^anx  Vsec2 x  -  1 

If  any  one  of  the  functions  is  given,  the  others  may  be 
found  from  these  formulas.  It  is  easier  in  general  to  find  first 
the  sine  and  cosine,  and  then  to  find  the  others. 

48.    Find  the  Unknown  Functions  in  the  Following  : 


1.  tana;=— -,  2:  being  in  the  fourth  quadrant.  Compute 
the  numerical  values  of  the  ratios 
by  the^  method  of  Art.  15,  and  then 
select  the  proper  signs  for  the  func- 
tions in  the  fourth  quadrant.     Thus  let 


0  =  3, 


sin  2:=  —  -, 
5 


4,    .  A  =  5 


cos  x  —  -\- 


5' 


cot  2;  =  --, 


sec  a:  =  H-  -, 
4 


cosec  a:  =  — -• 
o 


2.    tan  2:  =  2,  a;  being  in  the  third  quadrant.     Then 


sma: 


tana: 


cos  a: 


_1_ 

V5' 


Vl  +  tan2  X  V5'  Vi  +  tan2  x 

These  convenient  formulas  may  be  easily  remembered  from 
Fig.  35.     Knowing  sin  x  and  cos  a:,  we  have 

cota:  =  - =  4-^;  seca;  = =  _V5: 

tan  X  A  cos  x 


cosec  X 


sma;  z 


The  radicals  should  be  taken  with  the  double  sign. 


FUNCTIONS  OF  ONE  ANGLE.  49 

8.     cot  X  =  —  %  X  being  in  the  second  quadrant. 

•.   coseca;  =  ±  Vl  +  cot2a;=  +  V5;  sin  2:  = =H —  ; 

cosec  X  V6 

cos  x  =  ±  Vl  —  sin^a;  = ^  ;  tan  x  = ==  —  - ; 

V5  cot  X         2 

sec  a;  = =— -V5. 

cos  X  I 

4.    sec  x  =  —  -y-,  X  being  in  the  third  quadrant. 

.  •.  cos  X  = =  —  -—  ;  sin  a:  =  ±  V 1  —  cos'^^  = ; 

sec  a:  17  17 

,  sin  a:       ,  15         .  ,8  17 

tana;  = =-|-— -;   cota;  =  +-— ;   cosec  a;  = . 

cos  a;  8  15  15 

5.  sinx  =  —  I,  X  being  in  the  third  quadrant. 

.  ••  cos  X  =  —  f  ;  tan  x  =  +  | ;  cot  x  =  +  | ;  sec  x  =  —  | ;  cosec  id  =  —  | . 

6.  cosx  =  +  I,  X  being  in  the  fourth  quadrant. 

.-.  sinx  =  — |V6;  tanx  =  — ^VS;  cotx  =  — |\/5;  secx  =  +f; 
cosecx  =  — |V5. 

7.  tanx  =  —  y\,  X  being  in  the  second  quadrant. 

.  sin  X  =  +  t\  ;  cos  X  =  —  {|  ;  cot  x  =  —  -^  ;  sec  x  =  —  ^| ;  cosec x  =  +  ^?-. 

8.  cotx  =  +  j\,  X  being  in  the  third  quadrant. 

. '.  sin  X  =  —  II ;  cos  x  =  —  ^^. 

9.  sec  X  =  —  II,  X  being  in  the  second  quadrant. 

.  •.  cos  X  =  —  If ;  sin  X  =  +  y^y  ;  tan  x  =  —  -^-^. 

10.    cosec  X  =  —  V ,  X  being  in  the  fourth  quadrant. 


If  sin  J^: 
If  tan  J^; 

•.  sma 

^  =  -/t; 

;  CO 

.sx  =  +r_ 
where  s  = 

cos  J^=- 
where  s  = 

cosi5= 

;  tan  %■=- 

a^b^-c 
2        ' 

■^. 

11. 

W^ 

-6)(s- 
6c 

_^ 

show  that 

Ms-a)_ 
>       6c 
a  +  6  4-  c 
2        ' 

12. 

=# 

-a)(s- 
s(s  -  c) 

^ 

show  that 

ks-c), 

^         ^     ab 

13.    If  sec  6  =  a,  show  that  sin  6  is  imaginary  if  a  is  numerically  less  than 
unity. 


d  =  Vl  -  COS20  =a/1 ^  rrJl  -  1 

\        sec2^      ^        a2 


Smt/=V1  -COS''£'=A/l ^r-^-v/i  -—  =  — i. 

»2  /7 


CROCK.    TRIG.— 4  Vf^^^^^^'^^V 


'^     Of  THE 


50  PLANE   AND  ANALYTICAL   TRIGONOMETRY. 

14.    If  tan  6  =  a,  show  that  cosec  6  is  real  for  all  values  of  a. 

16.   If  cos  d  =  a,  show  that  cosec  6  is  imaginary  when  a  is  numerically 
greater  than  unity. 


49.  The  Signs  of  the  Functions  are  given  by  the  formulas 
of  Art.  46,  so  that  it  is  necessary  to  remember  only  that  the 
sine  is  positive  in  the  first  and  second  quadrants  and  the  cosine 
in  the  first  and  fourth.     Thus,  in  the  second  quadrant, 

sin  a;      -h                   ,          cos  a:      — 
tan  X  = =  —  =  —  ;  cot  X  = =  —  =  —  ; 

cos  a;      —  sma;      + 

1         +  1  +        ,        . 

sec  X  = =  —  =  —  ;  cosec  x  =  - —  =  —  =  + . 

cos  a;      —  sin  a;      + 


50.    Find  the  Values  of  the  Following  Expressions  : 

1.  versa:tan3T-l    ^^^^   tan  a:  =  4,  x   being   in   the  third 
secic 
quadrant.     Find  the  numerical  values  of  vers  x  and  sec  x,  and 
substitute. 

.-.  cosa;  = ^,  seca;  =  — VTf,  versa;=lH — . 

VlT  +  l.^      ^ 
Vr!  4VT7  +  4-V17  3VT7  +  4 


-VI7  -17  17       • 

2.     SIP  ^  sec  X     ^Yien  versx  =  |,  x  in  the  fourth  quadrant.         Ans.   +  16. 

cos  X  cosec  X 

8.   tan  a;  -  cot  X  ^^^^^  ^^^^^  x  =  -  V5,  x  in  the  third  quadrant.    Ans.   -  f . 
tan  X  +  cot  X 

when  cot  a;  =  —  ^,  a;  in  the  second  quadrant.    Ans.   —  2. 


cosec  X  4-  cos  x 

5.  sm  a;  +  tan  a;   ^^^^^^  sec  x  =  -  f ,  x  in  the  third  quadrant.       Ans.   +  ^%. 
cos  X  +  vers  x 

6.  sec  X  -  vers  x  ^^^^  cot  x  =  -  2,  x  in  the  second  quadrant, 
sec  X  4-  vers  x 

Ans    9  +  2V5^     29 -^  20V5 

7.  su^y^  +  t^"'^^    when  secx  =  -  |,  x  in  the  second  quadrant.  Ans.  ^Wz- 
cos-^  X  +  vers2  x 

8.  secx  +  sin  X  ^^^^  tan  x  =  2,  x  in  the  third  quadrant.       Ans.  -  ^ V5. 

1  —  cotx 


FUNCTIONS  OF  ONE  ANGLE.  51 

9.   cosecx  +  seca;  ^^^^  sec  a;  =  +  VIO,  x  in  the  fourth  quadrant. 

^°^^^°«^  Ans.   -20. 

10.  secx-coseca;  ^^^^  cotx  =  -  2,  x  in  the  second  quadrant.    Ana.  -  3. 
sec  X  +  cosec  x 

11.  versx- covers X  ^^^^  sinx  =  -  |,  x  in  the  fourth  quadrant. 

sec x- cosec X  ^,^^    _  ^^ 

51.    Change  the  Given  Expression  to  Another  containing  only 
One  Function : 

,    2  sec?  X  4-  sec^  x  tan^  x  —  sec*  x 


sec^  a;  —  1 


to  contain  only  cosec  x. 


It  is  best  generally  to  change  the  expression  to  another  con- 
taining only  sin  x  and  cos  x^  and  then  to  change  this  into  one 
containing  the  proper  function. 

2  sin2  X         1 


cos^  X     cos*  X      cos*  X  _2  cos^  X  +  sin^  x  —  1 
1  -|  cos2:c(l— cos^a;) 

COS^  2J 

2  —  2  sin^  2;  +  sin''^  a;  —  1  1  —  sin^  x  1  o 

; ! ; =  ; ^ =  — =  COSCC^  X. 

(1  —  sin^  x^  sin^  x         (1  —  sin^  x^  sin^  x     sin^  x 

to  contain  only  tan  x. 
vers  2:  —  covers  x 

sin^  a;  —  cos^  x  .        ,  ,        tan  2:        .  1 

=  sm  X  +  cos  x=  ± 


l-cosa;-l  +  sina:  Vl  +  tan^a;     VT+tan^^ 

where  the  signs  used  will  depend  upon  the  quadrant  of  x. 

The   true    result   is    ±  —  where   the   positive    sign 

Vl  +  tan^  X 
corresponds  to  x  in  the  first  or  fourth  quadrant,  and  the  nega- 
tive to  X  in  the  second  or  third. 
Use  radicals  as  little  as  possible. 

3.   1  —  2(1  —  covers  x)2  -\ ^~-^ — -  to  contain  only  cosx.    Ans.  cos*x. 

^  (1  +  t,an2  x)2  ^ 

4    sec xcosecx-4sinx cosx  ^^  ^^^^.^  ^^^^  ^.^ ^       ^,^^    (l-2sin2x)2^ 

sin  X  sec  x  sin^  x 

5.  (^  -  covers  x)^  cosec*  x  ^^  contain  only  tan  x.         ^ns.  tan2  x  +  tan*  x. 

(cosec- X  —  1)  cot-x 

«    sec2  X  —  sec2  x  sin*  x(l  +  cot^  x)  .  .   .        , 

6.   — ; 7^—^ ^  to  contain  only  cosec  x. 

sni2  x  cos2  X  . 

Ans.      *^^^^^*^ 


cosec2  X  —  1 


52  PLANE   AND   ANALYTICAL   TRIGONOMETRY. 

7.  tan2^sec2^-sin2^cos2^  to  contain  only  cot  ^.  Ans.  l+'^cot2g+3cot*g^ 

COt4^(l+COt2  0)2 

8.  ^    ~   ^"  ^^'  (cos*  X  —  sin*  x)  to  contain  only  sin  x.    Ans.  (1—2  sin2  x)*. 
(l  +  tan'-ix)--^^  ^  ^  \ 

9.  — sec2  a  sin2  a —  ^^  contain  only  cosec  a.  ^ns. 


(tan  a  +  2  cot  a)2  (2  cosec2  o  -  1)2 

-  10.      ^'^'^^^"'^    to  contain  only  sec  6.  Ans.   C^^^^'^--^)'. 

sin-  e  —  cos-^  d  sec2  ^  -  2 

-  11.   sec2^cosec2^+sec2^-cosec2^-l  ^^ ^^^^.^  ^^^ ^     ^^^^    cot2^+2 

tan2  d  -  cosec2  ^  +  1  1  -cot*  d 

52.  Solution  of  Trigonometric  Equations.  —  Transform  the 
given  equation  into  one  containing  only  a  single  function 
(usually  the  sine  or  cosine),  because  in  a  single  equation  we 
must  have  only  one  unknown  quantity.  Then  solve  the  equa- 
tion algebraically  for  this  function  as  the  unknown  quantity. 
The  corresponding  angle  may  then  be  found  from  the  tables. 
Test  the  angles  by  substitution  in  the  given  equation. 

I.  sin  ^  cos  ^  =  +  J. 

.-.  sin  eVl  -  sin2  (9  =  +  J  ;    .-.  sin2  (9(1  -  sin2  (9)  =  J  ; 
.-.  sin*(9-sin2(9  +  i  =  0;    .-.  sin2(9  -  1  =  0  ;    .-.  sin^  =  ±Vj. 

.-.  e  might  be  45°,  135°,  225°,  or  315°.  But  the  given  equa- 
tion shows  that  the  product  of  the  sine  and  cosine  must  be 
positive,  and  hence  that  they  must  have  the  same  sign.  Both 
the  sine  and  cosine  are  positive  in  the  first  quadrant,  and  nega- 
tive in  the  third,  but  they  have  contrary  signs  in  the  second 
and  fourth  quadrants.  Hence  the  only  admissible  values  of  6 
are  45°  and  225°. 

2.  tan  ^  sec  ^  =  -  \/2.  .-.  5  =  225°,  315°. 

3.  cosec  5  =  I  tan  5.  .-.  5  =  60^  300°. 

4.  tan 5  +  cot 5  =  2.  .-.  0  =  45°,  225°. 

6,  sec2 d  +  cosec2 6  =  4.  .:  d  =  45°,  135°,  225°,  315°. 

6.  sin5=±\/3vers5.  .-.  5  =  0°,  60°,  300°. 

7.  see  0  +  tan  5  =  ±  V3.  .-.  d  =  30°,  150°.               [300°,  330°. 

8.  sec2  0  +  cot2  5  =  J/ .  .-.  5  =  30°,  60°,  120°,  150°,  210°,  240°, 

9.  sinx=  +  V3cosx.  . •.  x  =  60°,  240°. 
'    10.  tan  X  =  -  2\/3  cos X.  .'.  x  =  240°,  300^ 

II.  sin  X  cos  X  =  -  1 V3.  .  •.  x  =  120°,  150°,  300°,  330° 

12.  sin  6  +  cosec  9  =  -^.  .:  $  =  210°,  330°. 

13.  3  sin  X  =  2  cos2  x.  .-.  x  =  30°,  150°. 

14.  sec  X  tan  X  =  +  2  V3.  .  •.  x  =  60°,  120°. 


FUNCTIONS  OF  ONE   ANGLE. 


68 


16.    sec  6  vers  6  =  \  —  tan  6. 

1     (l-C08^)=l-''"^ 


sin  ^  =  2  cos  ^  —  1 ; 


cos  6^  ^  cos^ 

.-.  sin2^  =  4cos2^-4cos^  +  l;  .-.  l-cos2^  =  4cos2^-4cos^  +  l; 
.  •.  5  cos2  ^  -  4  cos  ^  =  0  ;     .  •.  cos  ^  (5  cos  ^  -  4)  =  0  ; 
.  •.  cos  ^  =  0  and  5  cos  ^  —  4  =  0. 

(a)  cos  ^  =  0  gives  6  =  90°  or  270°.  These  values  are  re- 
jected for  reasons  involving  the  methods  of  the  Differential 
Calculus. 

(J)  cos  ^=f  gives  sin  ^=  ±  |,  since  this  value  of  the  cosine 
will  allow  the  angle  to  lie  either  in  the  first  or  in  the  fourth 
quadrant.     Transposing  in  the  original  equation,  we  have 

sec  6  vers  0  +  tan  ^  —  1  =  0, 

and  we  test  by  substitution.     For  0  in  the  first  quadrant,  we 
have 

showing  that  6  has  a  value  in  the  first  quadrant.     For  6  in 
the  fourth  quadrant,  we  have 

not  zero;  and  hence  6  does  not  have  a  value  in  the  fourth 
quadrant. 


16.  sin  X  tan  x  =  —  t%. 

17.  vers  x  =  2  covers  x. 

18.  sin  X  tan  x  =  2  cos  x. 

19.  sec  X  cosec  x  =  —  2. 

20.  cos  X  cot  X  =  —  f . 

21.  sin  X  cos  X  =  —  ^|. 

22.  tan  x  =  —  ^20  cos  x. 

23.  sec  X  +  tan  x  =  2. 


smx: 


5» 


cos  X  =  —  i  ;  quadrants  II.  and  III. 
•.  cos  X  =  I  or  —  1 ;  first  quadrant,  and  180°. 
•.  sin  X  =  ±  Vf ,  cos  X  =  ±  V-J  ;  four  quadrants. 
•.  sin  x  =  ±  ^\/2,  cos  X  =  T  1 V2  ;  135°  and  316°. 
•.  sin  X  =  —  f  ;  quadrants  III.  and  IV. 

•.  sin  X  =  ±  I  or  ±  f  ;  quadrants  II.  and  IV. 

2 
•.  sin  X  = :: ;  quadrants  III.  and  IV. 

V5 

•.  tan  X  =  +  f ;  first  quadrant. 


24.    sec  X  tan  2  a:  (1  —  2  cos  x)  =  0. 

The  values  of  x  are  found  by  placing  each  factor  equal  to 
zero,  and  solving  the  resulting  equations.     Hence  we  have 

sec  X  =  0,  tan  2  a;  =  0,  1  —  2  cos  x  =  0, 


64 


PLANE   AND   ANALYTICAL   TRIGONOMETRY. 


But  sec  a;  =  0  is  impossible;  tan  2  a;  =  0  gives  2  rr  =  0°  or  180°, 
and,  using  the  general  measures  of  the  angles,  x  =  0°,  90°, 
180°,  270°,  the  second  and  the  last  values  being  inadmissible. 
1  —  2  cos  a;  =  0  gives  cos  x  =  -J,  and  x  =  60°  or  300°. 

25.  tan  ^  a;  =  0. 

26.  vers  3  a;  =  0. 


27.  sinxcosx(l+2  cosx)=0. 

28.  cos  2  X  (3  -  4  cos^  x)  =  0. 

29.  (l+tana;)(l-2sinic)=0. 

30.  tan  X  =  —  2  sin  X. 

31.  sin  2x  vers3x  =  0. 


x  =  0°. 

X  =  0°,  120°,  240°. 

X  =  0°,  90°,  180°,  270°,  120°,  240°.  [330°. 

X  =  45°,  225°,  135°,  315°,  30°,  150°,  210°, 

X  =  45°,  225°,  30°,  150°. 
X  =  0°,  120°,  180°,  240°. 
X  =  0°,  90°,  180°,  120°,  240°,  270°. 


53.  The  Functions  of  an  Angle  Greater  than  360°  are  the 
same  as  those  of  the  angle  less  than  360°,  found  by  increasing 
or  diminishing  the  given  angle  by  some  multiple  of  360°  ;  for 
the  position  of  the  terminal  side  would  not  be  changed  by 
these  operations.     Thus 

tan  1010°  =  tan  (1010°  -  720°)  =  tan  290°  ; 
cos  ( -  835°)  =  cos  (  -  835°  +  720°)  =  cos  (  -  115°), 
or         cos  (-  835°)=  cos  (-  835°  +  1080°)=  cos  245°. 

54.  The  Functions  of  90°  ±  oc  and  of  270°  ±  a?  are  numeri- 
cally equal  to  the  cof unctions  of  x,  but  may  differ  from  them 
in  signs.      Let  the  arcs  UB,  EB,  KJ,  KM,  and  NP  be  equal, 


E 

T)^^ 

B 

/ 

\ 

.  / 

\ 

/ 

\ 

\  r 

^^. 

\ 

H 

i 

n 

r      \ 

/ 

V 

> 

/, 

J 

^■--^ 

_^^M 

JiV 


Fig.  87. 


K 

Fitt.  36. 


the  radii  CB  and  RP  each  being  unity.     Then  the  right  tri- 
angles PCB,  FOB,  LCJ,  LCM,  and  SRP  are  equal,  having 


FUNCTIONS  OF   ONE  ANGLE.  55 

the  same   hypotenuse    (unity)    and  the  angle  x  the  same  in 
each.      Therefore 

FB  =  CI=  LM=  DF=nC=  JL  =  SP, 
and  CF=IB  =  HD  =  LQ^  MI  =  JH  =  RS. 

.'.  sin(  ^()°-x)  =  IB  =  OF=RS  =  +  Go^x\ 


]    (4) 


cos(  W-x)=CI  =FB=SP  =  +  &mx.  )  ^^^ 

cos(  dO''+x)=CH:=FI)  =  -I)F*=-SP=-8inx.  ]  ^  ^ 
sin  (210°-x)  =  IfJ=  CL  =  -L0*=  -RS=  -  cos  a;; )  .^. 
008(270'' -x)=  Cir=LJ  = -JL*  =-SP=-sinx.]   ^  ^ 

sin  (210° +x}  =  lM=OL  = -LO*  = -RS= -cosx; 
cos  (270°  +  X)  =  CI  =  LM=  SP  =  +  sin  x. 

Thus    sin  100°=    sin  (90°  +  10°)=  +  cos  10°  ; 
cos  100°=    cos  (90°  +  10°)  =- sin  10°. 
sin  200°  =  sin  (270°  -  70°)  =  -  cos  70°  ; 
cos  200°  =  cos  (270°  -  70°)=  -  sin  70°. 
sin  300°  =  sin  (270°  +  30°)  =  -  cos  30°  ; 
cos  300°  =  cos  (270°  +  30°)  =  +  sin  30°. 

55.  The  Functions  of  i8o°  ±  y  and  of  360°  —  y  are  numeri- 
cally equal  to  the  same  functions  of  y,  but  may  differ  from 
them  in  signs.     From  Fig.  36, 

sin  (180°  -  z/)=  ^D  =  Zg  =  +  sin  y ; 


cos(180°  -y)=CH  =-  HO*  =  -CI  =  -  cos  ?/.  *  ^^^ 

8m(im° +y)=HJ  =  -JH*  =-Z5  =  -siny;)  .^^ 

cos(180°-hi/)=(7^  =-^(7*  =  -6Y  =-cos^/.  )  ^^ 

sin(360°-j/)  =  7ilZ=-if/*  =-Zg  =  -siny;)  ,„>. 

cos  (360° -I/)  =07"  =  +  cos2/.  )  ^^ 

Thus    sin  100°  =  sin  (180°  -  80°)  =  +  sin  80°  ; 
cos  100°  =  cos  (180°  -  80°)  =  -  cos  80°. 
sin  200°  =  sin  (180°  +  20°)=  -  sin  20°  ; 
cos  200°  =  cos  (180°  +  20°)  =  -  cos  20°. 
sin  300°  =  sin  (360°  -  60°)=  -  sin  60°  ; 
cos  300°  =  cos  (360°  -  60°)  =  +  cos  60 

*  See  Art.  2. 


\o 


66 


PLANE   AND  ANALYTICAL  TRIGONOMETRY. 


56.   The    Functions    of    a    Negative   Angle    are   numerically 
equal  to  the  same  functions  of  an  equal  positive  angle,  but  may 
differ  from  them  in  signs. 


B 

^-^ 

^ 

? 

^ 

B 

\ 

F 

/ 

\ 

H 

\ 

r 

^ 

t 

/ 

I 

/ 

\^ 

\ 

) 

^ 

L 

\ 

J 

J 

^--^ 

— 

M 

K 

Fig.  38. 


sin  (-?/)  =  IM=  -  MI""  =  -IB\ 

=  -  sin  «/ ; 
cos  (  —  1/)  =  CT=  +  cosy.  J 

Thus 
sin  (a:-180°)  =  sin  [-(180°-a:)] 

=  -  sin  (180°  -J)=-  sin  x. 
cos(a:-180°)  =  cos[-(180°-2;)] 

=  +  cos  (180°  —  x)=—  cos  X, 


(1) 


57.   Summary.  —  Using  the  equations  of  Art.  46, 


,  sin  a;       ,         cos  a; 

tan  X  = ,  cot  X  = , 

cos  X  ^  sin  X 


sec  re 


cos  a; 


cosec  X 


sin  a; 


and  the  results  of  Arts.  T>4:^35,  and  5Q,  we  have 


sin  (90°  —  a;)  =  + cosx; 

tan  (90°  -x)  =  +  cot  x  ; 

sec  (90°  —  x)  =  +  cosec  x  ; 

sin  (90°  +  x)  =  +  cos  a; ; 

tan  (90°  +  a:)  =  -  cot  X  ; 

sec  (90°  +  x)  =  —  cosec  x ; 
sin  (180°  -  x)  =  +smx;/\ 


QannSIE^xJ 


sec  (180°  —  x)  =  —  secx; 
sin  (180°  +  a;)  =  —  sinx  ; 
tan(180°  +  x)  =  +tanx; 
sec  (180°  +  x)  =  —  sec  X  ; 
sin  (270°  —  x)  =  —  cos  X  ; 
tan  (270°  -  x)  =  +  cot  x ; 
sec  (270°  —  x)  =  —  cosec  J  ; 
sin  (270°  +  x)  =  —  cos  X : 
tan(270°  +  x)  =  -cotx; 
sec  (270°  +  x)  =  +  cosec  x ; 
sin  (360°  -  x)  =  -  sin  x ; 
tan(360°-x)  =  -tanx; 
sec  (360°  —  X)  =  +  sec  X  ; 
sin  ( —  x)  =  —  sin  x  ; 
tan  (  —  x)  =  —  tan  x  ; 
sec(— x)  =  +  secx; 


cos   (90°  —  x)  =  +  sin  X ; 

cot    (90°  -  x)  =  +  tan  X  ; 
cosec   (90°  —  x)  =  +  sec  x. 

cos   (90°  +  x)  =  —  sin  X ; 

cot  (90°  +  x)  =  - tanx; 
cosec  (90°  +  x)  =  +  sec  x. 

cos  (180°  —  x)  =  —  cos  X ; 

cot(180°-x)  =  -cotx; 
cosec  (180°  —  x)  =  4-  cosec  x. 

cos  (180°  +  x)  =  —  cosx  ; 

cot(180°  +  x)  =  +  cotx; 
cosec  (180°  +  x)  =  —  cosec  x. 

cos  (270°  —  x)  =,^_sinjc  ; 

cot  (270°  -  X)  =  +  tan  x  ; 
cosec  (270°  —  x)  =  —  sec  x. 

cos  (270°  +  x)  =  +  sin  x  ; 

cot  (270°  +  x)  =  -  tan  X  ; 
cosec  (270°  -|-  x)  =  —  sec  x. 

cos  (360°  -  x)  =  +  cos  X  ; 

cot(360°-x)  =  -cotx; 
cosec  (360°  —  x)  =  —  cosec  x. 

cos  (  —  x)  =  +  cos  X  ; 

cot  (—  x)  =  —  cotx  ; 
cosec  (  —  X )  —  —  cosec  x. 

*  See  Art.  2. 


(1) 


(2) 


(3) 


(4) 


(5) 


(6) 


(7) 


(8) 


FUNCTIONS  OF  ONE  ANGLE. 


67 


These  formulas  may  be  remembered  from  the  three  facts  : 

(a)  Whenever  the  angle  is  90°  ±  x,  or  270°  ±  x,  the  func- 
tions of  the  angle  are  numerically  equal  to  the  corresponding 
(7of  unctions  of  x. 

(b)  Whenever  the  angle  is  180°  ±  x,  360°  -  a;,  or  -  x,  the 
functions  of  the  angle  are  numerically  equal  to  the  same 
functions  of  x. 

((?)  The  sign  to  be  placed  before  the  function  of  x  is  that  of 
the  original  function  when  x  is  less  than  90°.     Thus 

sin  (270°  +  x)  =^-.  cos  x, 

since,  when  x  <  90°,  270°  +  x  will  be  in  the  fourth  quadrant, 
and  sin  (270°  +  x)  will  therefore  be  negative. 

58.   General  Method  of  Proof.  —  In  Arts.   54,   55,  and  56, 
both  x  and  y  were  less  than  90°,  but  the  formulas  in  Art.  57 
are  true  for  all  values  of  x.     Sup- 
pose, for  example,  tbat  we  wish  to 
prove    the    formulas    for    270°  4-  x 
when  X  is  in  the  fourth  quadrant, 
that    is,   when    x    is    between   270° 
and  360°.     Let  KAUaJ=x;    then 
AEaJKAEaJ=  270°  +  x.         Let 
AEGKJ'^x.      Then   in   the   right 
triangles  JCL  and  J'  CL'  the  angles 
JCL  and  J'  CL'  are  equal,  each  be- 
ing 360°  —  X  ;  therefore  the  triangles 
are  equal,  and  CL  =  CL'  and  LJ=  L'J'  numerically, 
braically  CL  =  -  CL'  and  LJ=  +  L'J'. 

.-.  sin(270°  +  2:)=5'J^=  CX  =  - OX'   =-cosa:; 
cos  (270°  ■\-x)=  CH=  LJ  =  +  L'J'  =  +  sin  a;. 

EXAMPLES. 
1.   From  the  preceding  equations  prove  that 


Alge- 


(a)  tan  (-  1200°)  =  cot  30°. 
(&)  sec  1000°  =  cosec  10°. 
(c)  cos  (  -  890°)  =  -  cos  10°. 
{d)  cot  1700°  =  cot  80°. 
(e)  cosec  (  -  1235°)  =  -  sec  66°. 
(/)  sin  1340°  =  -  cos  10°. 


(gr)  sin  (  -  3000°)  =  -  cos  30°. 
(/i)  cos  1300°  = -cos 40°. 
(0  tan 3200°  = -tan 40°. 
( j)  cot  ( -  1300°)  =  -  cot  40°. 
(k)  sec  (-  2900°)  =  +  sec  20°. 
(0  cosec  2420°  =  -  sec  10°. 


58  PLANE   AND  ANALYTICAL  TRIGONOMETRY. 

2.   If  tan  ^  =  -  cot  140°,  find  the  two  values  of  6  less  than  360° 
. •.  tan 5  =  -  cot  (90°  +  50°)  -  +  tan 50°.  .-.  0  =  50^  230°. 

8.  Find  the  values  of  6  in  the  following  equations : 

(a)        sin  ^  =  +  cos  220°.  .  •.  230°,  310°. 

(6)        sin^  =  +  cos310°.  .-.  40°,  140°. 

(c)  sin^  =  -cos210°.  .-.  60°,  120°. 

(d)  sin2  d  =  +  cos2  200°.       .  •.  70°,  110°,  250°,  290°. 

(e)  cos  ^  =  +  sin  150°.  .  •.  60°,  300°. 
(/)  cos  ^  =  +  sin  250°.  .  •.  160°,  200°. 
{g)  QOsd  =  -  sin  170°.  .-.  100°,  260°. 
Qi)  cos  ^  =  -  sin  275°.  .-.  5°,  355°. 

(0  cos2  ^  :=:  +  sin2  100°.  .'.  10°,  350°,  170°,  190°. 

(j)  tan  ^  =  +  cot  100°.  .  •.  170°,  350°. 

(A:)  tan  ^  =  +  cot  200°.  .  •.  70°,  250°. 

(0  tan^  =  -cot230°.  .-.  140°,  320° 

(?n)  cot  0  =  +  tan  260°.  .  •.  10°,  190°. 

(n)  cot^ii:+tan345°.  .-.  105°,  285°. 

(o)  cot  (?  =  -  tan  245°.  .  •.  155°,  335° 

{p)  cot  ^  =  -  tan  305°.  .-.  35°,  216°. 

(g)  sec  ^  =  -  cosec  100°.  .-.  170°,  190°. 

(r)  sec  0  =  +  cosec  130°.  .  •.  40°,  320°. 

(s)  sec  ^  =  +  cosec  310°  .  •.  140°,  220°. 

(0  cosec  ^  =  +  sec  315°.  .  •.  45°,  135°. 

(It)  cosec  ^  =:  +  sec  2.30°.  .  •.  220°,  320°. 

(v)  cosec  ^  =  -  sec  185°.  .  •.  85°,  95°. 

{w)  cosec  ^  =  -  sec  335°.  .  •.  245°,  295°. 

(x)  cosec2  0  zr  +  sec2  250°.  .-.  20°,  160°,  200°,  340°. 

{y)  SQCd  =  -  cosec 290°.  . •.  20°,  340°. 

{z)  sin  ^  =  -  cos  300°.  .  •.  210°,  330°. 

4.  cos  0  =  sin  2  6.     Show  that  one  value  of  0  is  30°. 

5.  tan  n  0  =  -  cot  120°.     Show  that  one  value  of  6  is  30°  ^  n. 

6.  sec3e  =  cosec  (n- 1)^.     Show  that  one  value  of  ^  is  90° -^(n  +  2). 

7.  If  cot  309°  =  -  j%,  find  sin  219°. 

sin  219°  =  sin  (180°  +  39°)  =  -  sin  39°. 

But  cot  309°  =  cot  (270°  +  39°)  =  -  tan  39°  ;     .-.  tan  39°  =  +  j%. 

...  sin39°  =  -^:^|?L==-4=  =  -^;     .'.  sin219°  =-- 4< 
Vl  +tan2  39°      Vl64      Vil  V41 

8.  If  sin  21 7°  =  -  j%,  prove  that  tan  127°  =  -  f- 

9.  If  cos  125°  =  -  a,  prove  that  tan  325°  = _^ - 

vl  —  a2 


FUNCTIONS  OF  ONE   ANGLE.  59 

10.   If  cot  260°  =  +  a,  prove  that  cos  350°  =  +       ^ 


vTTa2 
11.   If  sec  340°  =  +  a,  prove  that  sin  110°  =  - ,  and  tan  110°  = — 


12.  If  cos  300°  =  +  «,  prove  that  cot  120°  = ^ 

13.  If  sin  116^  =  + a,  prove  that  tan  205°  sec  245°  _  .  VTIT^^ 


cosec  385°        a 

14.  If  cos  200°  =  -  m,  prove  that  tan  110°  cosec  250°  cot  290°  = 

m 


15.  If  cosec  185°=  -m,  prove  that  tan  355°  tan  275°  cos  175°  =  -  ^^ — -. 

m 

16.  Show  that  cot  ^  (  -  x  -  540°)  =  tan  J  x. 

cot  K- X  -  540°)  =  cot  (- ^  X  -  90°) 

=  cot  [ -  (90°  +  1  X)]  =  -  cot  (90°  +  i  sc)  =  +  tan  ^ SB. 

17.  Show  that  sin  (y  —  90°)  =  —  cos  y. 

18.  Show  that  sin  (y  -  180°)  =  -  sin  y.  - 

19.  Show  that  cos  (y  —  270°)  =  —  sin  y. 

20.  Show  that  sec  (—  x  —  540°)  =  —  sec  x. 

21.  Show  that  tan  (y  -  360°)  =  +  tan  y. 

22.  Show  that  cos  |(x  -  270°)  =  +  sin  | x.    [Note  that  270°  in  the  parenthe- 
sis is  to  be  multiplied  by  i.] 

23.  Show  that  cos  |( -  810°  +  a  -  6)  =  -  sin  ^(a  -  6). 

24.  Show  that  cosec  |(x  -  360°)  =  -  sec  ^  x. 

25.  Show  that  sec  ^  (  -  900°  -  x)  =  -  sec  ^  x. 

26.  Show  that  tan  |(360°  +  a  -  &)  =  +  tan  ^(a  -  6). 

27.  Show  that  cos  (180°  —  x)  is  equal  to  the  sine  of  the  complementary 
angle. 

Complement  =  90°  -  (180°  -  x)  =  -  (90°  -  x)  ;     sin  [-  (90°  -  x)]  = 

-  sin  (90°  -x)=  -  cos  x.     But  cos  (180°  -  x)  =  -  cos  x.      .  •.  cos  (180  -  x)  = 
sin  [90°- (180°- X)].  q.  e.  d. 

28.  Show  that  cosec  (270°  —  x)  equals  the  secant  of  the  complementary 
angle. 

29.  Show  that  tan  (180°  +  x)  equals  the  cotangent  of  the  complementary 
angle. 

30.  Show  that  sec  (270°  +  x)  equals  the  cosecant  of  the  complementary 
angle. 

31.  Show  that  cos  (90° +x)  equals  the  sine  of  the  complementary  angle. 

32.  Show  that  cot  (360°— x)  equals  the  tangent  of  the  complementary  angle. 

33.  Show  that  tan  (270°+ a;)  is  equal  to  the  negative  of  the  tangent  of  the 
supplementary  angle. 

Supplement  =  180°  -  (270°  +  x)  =  -  (90°  +  x)  ;      tan  [-  (90°  +  x)]  = 

-  tan  (90°  +  x)  =  +  cot  x.     But  tan  (270°  +  x)  =  -  cot  x.     .  •.  tan  (270°  +  x)  = 

-  tan  [180°  -  (270°  +  x)].  q.  e.  d. 

34.  Show  that  cosec  (180°  +  x)  is  equal  to  the  cosecant  of  the  supple- 
mentary angle. 


60  PLANE   AND   ANALYTICAL   TRIGONOMETRY. 

35.  Show  that  sin  (360°  —  x)  is  equal  to  the  sine  of  the  supplementary 
angle. 

36.  Show  that  sec  (90°  +  x)  is  equal  to  the  negative  of  the  secant  of  the 
supplementary  angle. 

37.  Show  that  cos  (270°— cc)  is  equal  to  the  negative  of  the  cosine  of  the 
supplementary  angle. 

38.  Show  that  cot  (270°  +  x)  is  equal  to  the  negative  of  the  cotangent  of 
the  supplementary  angle. 

59.  The  Trigonometric  Tables.  —  The  relations  shown  in 
Arts.  53  and  51  enable  us  to  find  the  functions  of  any  angle, 
although  the  tables  contain  only  the  sines,  cosines,  tangents, 
and  cotangents  of  angles  less  than  45°.     For,  since 

sin  (90°  —  x)  =  cos  x,  cos  (90°  —  x^  =  sin  x, 
tan  (90°  -x)  =  cot  x,  cot  (90°  -  x)  =  tan  x, 

the  tables  are  immediately  extended  to  90°  by  writing  the 
proper  degrees  and  minutes  at  the  bottom  and  on  the  right 
of  the  page  respectively. 

Then,  since  the  value  of  any  function  of  an  angle  greater 
than  90°  can  be  found  in  terms  of  a  function  of  an  angle  less 
than  90°,  we  can  find  the  numerical  value  of  the  function  from 
the  tables. 

1.   Find  from  the  tables  the  logarithmic  functions  of  580°  42'.4. 

580°  42' A  =  300°  +  220°  42'.4. 
.-.  sin  580°  42'.4  =  sin  220°  42'.4  =  sin  (180°  +  40°  42'.4)  =  -  sin  40°  42'.4 ; 

.-.  log  sin  580°  42'.4  =  9.81437  n. 


cos  580°  42'.4  =  -  cos  40°  42'.4 
tan580°42'.4  =  +  tan  40°  42'.4 
cot  580°  42'.4  =  +  cot  40°  42'.4 


.-.  log  cos  580°  42 '.4  =  9.87971  n. 
.'.  log  tan  580°  42'.4  =  9.93467. 
.-.  los  cot  580°  42'.4  =  0.06533. 


2.   Find  from  the  tables  the  logarithmic  functions  of  the  following  angles : 


Angle. 

log  sin. 

log  COS. 

log  tan. 

log  cot. 

499°  29'.7. 

9.81258. 

9.88102  w. 

9.93158  w. 

0.06842  n. 

597°    S'.B. 

9.92427  w. 

9. 73449  «. 

0.18978. 

9.81022. 

689°  27'.6. 

9. 70598  w. 

9.93514. 

9.77084  w. 

0.22916  n. 

3.  sin  b  =  tan 250°  15'.5  cot 278°  17'.3  ;  find  6  =  203°  57'.0  or  336°  3'.0. 

4.  cos  /3  =  cos  149°  27'.6  sin  216°  44'.0  ;  find  8  =    58°  59^7  or  301°  0'.3. 

6.  tana  =  sin 319°  52'.0  --  cot 254°  30'.2  ;  find  a  =  113°  16'.5  or  293°  16'.5. 

6.  cotc  =  cos216°44'.0-v-tan329°27'.6;  find  c  =    36°21'.6  or  216°21'.6. 


FUNCTIONS  OF  ONE  ANGLE.  61 

60.   Transform  the  First  Member  into  the  Second   in  the 

following  examples.  Usually  it  is  best  to  change  the  given 
expression  into  one  containing  the  sine  and  cosine,  and  then  to 
change  this  into  the  required  form.  Any  operation  is  admis- 
sible that  does  not  change  the  value  of  the  expression.  Use 
radicals  only  when  unavoidable.  If  the  expression  is  factored, 
it  is  often  advantageous  to  reduce  each  factor  separately,  not 
multiplying  until  it  becomes  necessary. 

tan  X  —  sin  x         sec  a? 


sin^  X  1  -f  cos  X 

sin  a; 


—  sin  X 


tan  X  —  ^\nx     cos  x  _  sin  a;  (1  —  cos  a;)  _  1  —  cos  x 

sin^  x        ~~        sin^  x  cos  x  sin^  x  cos  x  sin^  x 

1  —  cos  X  1  sec  X 


cos  x(l  —  cos^  x^      cos  x(l  -\-  cos  a;)      1  +  cos  x 

2.  COS  X  tan  a;  +  sin  x  cot  x  =  sin  oj  +  cos  x. 

3.  (2  —  vers  x)  vers  x  =  sin^  x. 

.  COSX  f„„  « 

4.   =  tan  X. 

sin  X  cot2  X 

5.  (tan  X  +  cot  x)  sin  x  cos  x  =  1. 

6.  (sec^x- l)(cosec2x  — 1)  =  1. 

7.  sec  X  cosec  x  (cos^  x  —  sin^  x)  =  cot  x  —  tan  x. 

8.  (sin  X  +  cos  x)  (tan  x  +  cot  x)  =  sec  x  +  cosec  x. 

9.  cot  X  +     ^^^^     =  cosec  x. 

1  +  cos  X 

10.  sin  X  (sec  x  +  cosec  x)  —  cos  x  (sec  x  —  cosec  x)  =  sec  x  cosec  x. 

11.  (cosec  X- cot  x)2  =  ^  -cosx^ 

1  +  cos  X 

12.  (1  +  tan2  x)  (1  -  cot'2  x)  =  sec^  x  —  cosec^  x. 

13.  tan  X  -  cot  X  _  — 2 ^      [First  change  to  an  expression  containing 

tan  X  +  cot  X     cosec^  x 

only  sin  x,  the  reciprocal  of  cosec  x.] 

14.  sec2  X  cosec2  x  —  2  =  tan^  x  +  cot^  x.     [Substitute  for  sec  x  and  cosec  x 
their  values  in  terms  of  tan  x  and  cot  x  respectively.  ] 

,K    tan  a  +  tan /3     .„„     ♦„„  o 

15.    ' —  tan  a  tan  p. 

cot  a  +  cot  j3 

16.  cot  X  -  sec  X  cosec  x  (1  -  2  sin^  x)  =  tan  x.     [The  expression  reduces  to 

sill  X  H- cos  X.] 


62  PLANE   AND   ANALYTICAL   TRFGONOMETRY. 

17.  cosec  X  (sec  a;  —  1)  —  cot  x  (1  —  cos  x)  =  tan  x  —  sin  x.     [Factor  a^  sood 
as  possible,  and  reduce  each  factor  separately.  ] 

18.  vers  x  (sec  x  +  1 )  +  covers  x  (cosec  x  +  1)  =  sin  x  tan  x  +  cos  x  cot  x. 

j3^   vers  x  (1  +  sec  x)  ^  covers  x  (1  +  cosec  x)  ^  ^^^  ^  ^^^^^  ^ 
sin  X  cos  X 

20.  sin2  X  (tan2  x  -  1)  +  cos^  x  (cof-^  x  -  1)  =  0  -  2  cos'^  x)^  seC^  x^ 

tan'^  X 

21.  sec  X  cosec  x [vers  x(vers  x  —  2)  —covers  x(covers  x— 2)]  =cot  x— tan  x. 

22.  cos*  X  —  sin*  x  =  cos  x  ( 1  —  tan  x)  (sin  x  +  cos  x) . 

^„     vers X  (1  +  cos  x)— covers X  (1  +  sin  x)      tan*x  — tan^x      ^^, 

23.   ^^ r 5 ^ -  — [Change  to 

sec-*  X  cosec-^  x  sec  ^  x  to 

an  expression  containing  only  sin  x  and  cos  x,  and  then  substitute  their  values 
in  terms  of  tan  x.  ] 

g,     sec'^  X  sin^^  x  —  cosec'^  x  +  cosec'^  x  cos^  x  _   .  o  ^ 
sec2  X  sin2  x  —  cosec^  x  cos^  x 

„c    *     o  -9  -        (sec2x  +  l)(sec2x- 1)2 

25.   tan2  x  -  sin2  x  cos^  x  =  -^^ ^-^^ —- 

sec*  X 

2g    cosxcotx-sinxtanx^l_^3.^^^^3^ 
cosec  X  —  sec  X 

2„     (sec  X  4-  cosec  x)2  _  (1  +  tan  x)^ 
tan  X  +  cot  X  tan  x 

2g_   2  +  5il^i^il£2!i£  =  sec2x  +  cosec2x. 
sin2  X  cos2  X 

gg    sec  X  +  cosec  x  _  tan  x  +  1  _  1  +  cot  x 
sec  X  —  cosec  x     tan  x  —  1     1  —  cot  x 

„Q    sin  X  —  tan2  x  covers  x  sin*  x 


cosec  X  cot2  X  (1  +  sin  x)  cos2  x 

81.  sec2  X  cosec2  x  [vers  x  (vers  x  —  2)  —  covers  x  (covers  x  —  2)  ] 

=  cot2  X  —  tan2  X. 

82.  tan  x  +  cot  x  =  ^^^^  ^  "^  cosec2  j;^      ^^^    .^    admissible    to    divide    both 

sec  X  cosec  x 
numerator  and  denominator  by  sin2  x  cos2  x.] 

88.   tan2  a  tan2 /3  -  1  =  ?^5!^^^^  =  ^H^^f=^. 
cos2  o  cos2  /3  C0S2  a  cos2  /3 

g^     1  -  tan2  g  tan2  /3  _  cos2  a  -  sin2  j3  _  cos2  ^  -  sin2  g 
tan2  g  tan2  /3     ~    sin2  ^  sin2  /3    ~    sin2  g  sin2  ^ 

85.   sin2  X  tan2  x  +  cos2  x  cot^  x  =  tan2  x  +  cot2  x  —  1. 

36.  sin2  X  tan  x  +  cos2  x  cot  x  +  2  sin  x  cos  x  =  sec  x  cosec  x. 

37.  sec*  X  +  tan*  x  =  1  +  2  sec2  x  tan2  x.     [It  is  admissible  to  add  and  sub- 
tract 2  sec2  X  tan2  x.  ] 

38.  (r  cos  0)2  +  (r  sin  0  sin  0)2  +  (r  sin  </>  cos  0)2  =  r2. 

.  •.  r2  cos2  0  +  r2  sin2  0  (sin2  e  +  cos2  0)  =  r2  (cos2  0  +  sin2  0)  =r2, 
since  sin2  x  +  cos^  x  =  1. 


FUNCTIONS  OF  ONE  ANGLE.  63 

89.  (2  r  sin  a  cos  o)*  +  r^  (cos2  a  -  sin^  o)2  =  r^. 

40.  (a  sin  7)2  +  (a  cos  7  sin  5)^  -|-  (a  cos  7  cos  5)2  =  a^. 

41.  (cos  a  cos  &  —  sin  a  sin  6)2  +  (sin  a  cos  6  +  cos  a  sin  6)2  =  1. 

42.  (cos  a  cos  6  +  sin  a  sin  6)2  +  (sin  a  cos  b  —  cos  a  sin  6)2  =  l. 

43.  (x  cos  ^  -  y  sin  ^)2  +  (x  sin  ^  +  ?/  cos  dy  =  x2  +  y2. 

^    1 4  tan2  X      _  , 

(cos2  X  -  sin2  a;)2     (1  -  tan2  x)2  ~   * 

45.  1  (l-tan2x)2^^ 
4  sin2  X  cos2  x         4  tan2  x 

46.  (3  sin  a  cos2  a  -  sin^  a)2  +  (cos^  a  -  3  sin2  a  cos  a)2  =  1. 

47.  x^-hy^  +  s^  =  ?-2  when 

X  =  r  cos  a  cos  /3  +  r  cos  i  sin  a  sin  /3, 
y  =  r  cos  I  cos  a  sin  /3  —  r  sin  a  cos  /3, 
2  =  r  sin  t  sin  /S. 


CHAPTER  V. 


RELATIONS  BETWEEN  FUNCTIONS  OF  SEVERAL  ANGLES. 


61.   Sine  and  Cosine  of  the  Sum  of  Two  Angles.  —  Let  x  and 

y  be  the  angles,  each,  as  well  as  their  sum,  being  less  than  90°. 

QQ  \s>  perpendicular  to  OP,  BC  and 
BQ  are  perpendicular  to  OA^  and 
EC  is  parallel  to  OA^  the  radius  of 
the  circle  being  unity.     Then 

x-\-y  =  AOQ, 

the  angle  EQC=x^  06'=  cos  i/,  and 
CQ  =  sin  y. 

sin  (x-^y')=DQ  =  BO-\-  EQ 

=  OC^mBOC+  OQ  cos EQC 

=  cos  y  sin  x  +  sin  y  cos  x, 

or  sin  (a?  +  y)  =  sin  a?  cos  ?/  +  cos  a?  sin  y.  (1) 

cos  (a: +  2/)=  ^J^=  OB-EO=  OOcosBOO-  CQsmEQO 
=  cos  y  COS  X  —  sin  ^  sin  x, 
cos  (ac  +  y)  =  cos  a?  cos  y  -  sin  a?  sin  y.  (2) 


or 


1.  sin  90°  =  sin  (60°  +  80°)  =  sin  60°  cos  30°  +  cos  60"  sin  30° 

^\/8     V3  ,  1.  1^1 
2*2        2*2 

2.  cos  90°  =  cos  (60°  +  30°)  =  cos  60°  cos  30°  -  sin  60°  sin  30° 

=  1.  V3   V3  ,  1  ^  0 

2*2    2*2 

3.  If    sin  a  =  f ,  and   sin  /3  =  y\,   find   sin  (a  +  /3)    and  cos  (a  +  /3)    when 
a  <  90°,  and  /3  <  90° 


A71S.  sin  (a  +  iS)  =  |f ,  cos  (a  +  /S)  =  |f 


64 


J'UNCTIONS  OF  SEVERAL   ANGLES. 


66 


4.   If   tana  =  f,  and  tan/3  =  ,'j,   find   sin  (o  + /3)    and   cos(o  +  /3)   when 

a  <  90°,  and  /3  <  90°. 

Ans.  sin  (a  +  P)=  *,  cos  (o  +  /3)  =  g. 

Note.  —  At  a  point  A  the  angle  of  elevation  DAB  to  the  top  of  a  vertical 
wall    is  o,  and   the   angle   of    depression 
CAD  to  its  base   is  /3.     Find  the  height 
CB  of  the  wrall,  the  horizontal  distance 
AD  being  a  feet. 

CB=  CD  -\-  DB  =  a  tan  /3  +  a  tan  a 

=  a(tana  + tan/3).  (3) 

^^/sina^sin^X 
\cosa      COSjS/ 
_    sin  g  cos  /3  +  cos  a  sin  /3 


cos  a  cos  /3 

^sin  (g  +  /3) 
cos  g  cos  /3 


Fig.  41. 


(4) 


Eq.  (3)  w^ould  be  solved  by  the  use  of  the  natural  functions,  while  (4)  is 
adapted  to  logarithmic  computation. 

62.  Sine  and  Cosine  of  the  Difference  of  Two  Angles.  —  Let 
X  and  y  be  the  angles,  each  being  less  than  90°  and  x  being 
greater  than  t/.  QC  is  perpendicu- 
lar to  OP,  BO  and  DQ  are  perpen- 
dicular to  OA,  and  EQ  is  parallel 
to  OA,  the  radius  of  the  circle 
being  unity.  Then  x  —  y  =  AOQ, 
EOQ=x,  0(7=cos?/,..and  C§=siny. 

sin  (x-y)=DQ=  BO-  EO 

=  0Os'mB0O-  OQcosEOQ 

=  cos  ?/  sin  X  —  sin  «/  cos  x, 

or  siii(x-2/)  =  siu£ccos2/-cosa5sin2/.  (1) 

cos  (x  -  y)=  OB  =  OB  +  EQ  =  00  cos  BOO  +  OQ  sin  EOQ 

=  cos  1/  cos  X  +  sin  z/  sin  x, 

or  cos  (pc-y)=  cos  a?  cos  y  +  sin  x  sin  y,  (2) 

In  this  proof  we  have  assumed  that  x  is  the  greater  angle, 
but  (1)  and  (2)  are  true  when  y  is  greater  than  x.  To  prove 
this,  let  y8  be  greater  than  «.     Then 

sin  (a  —  /3)  =  sin  [  —  (/8  —  «)]  =  —  sin  (yS  —  a), 

CROCK.   TRIG.  —  5 


66  PLANE  AND  ANALYTICAL  TRIGONOMETRY, 

and,  developing  sin  (^  —  a)  by  (1), 

sin  (rt  —  yS)  =  —  (siny8  cos  a  —  cos  ^  sin  a) 

ss  sin  a  cos  /S  —  cos  a  sin  ^. 
Also     cos  (a  —  /3)  =  cos  [  —  (/S  —  «)]  =  cos  (/S  —  a) 

=  cos  )Q  cos  a  H-  sin  /3  sin  a. 


Q.E.D. 


Q.E.D. 


1.   sin  30°  =  sin  (90°  -  60°)  =  sin  90°  cos  60°  -  cos  90°  sin  60° 


1.1-0.^ 


2.  cos  30°  =  cos  (90°  -  60°)  =  cos  90°  cos  60°  +  sin  90°  sin  60° 

=  0.1+1.^  =  ^. 

2  2         2 

3.  If  sin  a  =  j^j,  and  sin  /3  =  :f\,  find  sin  (a  -  /3)    and   cos  (a  -  /3)   when 
o<90°,and/3<90°. 

Ans.  sin  (a  -  /S)  =  ^^i  \  cos  (a  -  /3)  =  |f  f . 

4.  If  tan  a  =  f ,   and  tan/3  =  f,   find   sin  (a  —  /3)    and   cos  (a  —  /3)    when 
a  <  90°,  and  /3  <  90.° 

Ans.  sin  (a  -  /3)  =  ^V  J  cos  (a  -  /3)  =  f  f . 

Note.  —  At  a  pointy^  on  a  horizontal  plane,  the  angle  CAD  to  the  top  of 

a  crag  is  7,  and  a  feet  farther  away  in  the  same 

vertical  plane  (at  5),  the  angle  CBD  is  7'. 

^,,„, .,,       Find^C  =  a;. 

Ci>  =  ^C  tan  7  =  a;  tan  7. 

CD  =  BCid^ny'  =(a  +  a;)  tan  7'. 

.  •.  X  tan  7  =  (a  +  a;)  tan  7'. 


Fig.  43. 


^  _       q  tan  7^ 
tan  7  —  tan  7' 

osiny  C0S7 


g  sin  7^  cos  7 

sin  7  cos  7' —  cos  7  sin  7'      sin  (7  — 7') 


(3) 

(4) 


Eq.  (3)  would  be  solved  by  the  use  of  the  natural  functions,  while  (4) 
is  adapted  to  logarithmic  computation. 

63.  General  Proof  of  the  Addition  Formulas.  — These  formulas  were 
shown  in  Art.  61  to  be  true  when  a;,  y^  and  x  +  y  were  each  less  than  90°.  That 
they  are  true  for  all  values  of  these  angles  may  be  shown  by  proving  the  special 
cases  separately.     Let  us  consider  first  the  case  when 


»<  90°,  y  < 90°,  X  +  2/ > 90°  and  <  180^^ 


FUNCTIONS  OF  SEVERAL   ANGLES.  67 

Let         x  =  90°-a,  y  =  90°-p;   .-.  x  +  y  =  180°  -(a  +  /3). 
.-.  a  =  90°  -  x,  )8  =  90''  -  y,  a  +  i8  =  180°  -  (x  +  j/). 
.-.  a  <  90°,  /3  <  90°,  a  +  /3  <  90°,  since  a;  +  y  >  90°. 
Then  sin  (a  +  /3)  may  be  developed  by  (1),  Art.  61,  since  the  conditions  of  that 
article  are  satisfied.     But 

sin  (x  +  y)  =  sin  [180°  —  (a  +  /3)]  =  sin  (a  -f  |3)  =  sin  a  cos  /3  +  cos  a  sin  /3 
=  sin  (90°  -  x)  cos  (90°  -  y)  +  cos  (90°  -  x)  sin  (90°  -  y) 
=  cos X  sin  2/ +  sin  05  cos  2/.  q.e.d. 

Also  cos  (x  +  y)~  cos  [180°  -  (a  +  /3)]  =  -  cos  (a  +  ^)  =  -  cos  a  cos  j8  +  sin  a  sin  |3 
=  -  cos  (90°  -  x)  cos  (90°  -y)  +  sin  (90°  -  x)  sin  (90°  -  y) 
=  —  sin  X  sin y  4- cos  X  cos  y.  q.e.d. 

Hence  the  formulas  are  true  for  x  <  90°,  y  <  90°,  x  +  y  <  180°. 

To  illustrate  the  proof  for  any  special  case,  let  us  take  x  in  the  second  and 
y  in  the  fourth  quadrant.  Place  x  =  90°  +  a,  and  y  =  270°  +  /3,  so  that  a  and  /3 
are  each  less  than  90°.    Then 

sin  (x  +  y)=  sin  [360°  +  (a  +  /3)]  =  sin  (a  +  jS)  =  sin  a  cos  /3  +  cos  a  sin  /3 
=;  sin  (x  -  90°)  cos  (y  -  270°)  +  cos  (x  -  90°)  sin  (y  -  270°) 
=  (— cos  x)(— sin  ?/)  +  sin X  cos  y 

=  cos  X  sin  y  +  sin  x  cos  y.  q.e.d. 

Let  the  student  prove  the  addition  formulas  in  the  following  cases : 

1.  X  in  the  first,  and  y  in  the  third  quadrant. 

2.  X  in  the  second,  and  y  in  the  second  quadrant. 

3.  X  in  the  second,  and  y  in  the  third  quadrant. 

4.  X  in  the  third,  and  y  in  the  third  quadrant. 
6.   X  in  the  third,  and  y  in  the  fourth  quadrant. 
6.    X  in  the  fourth,  and  y  in  the  fourth  quadrant. 

64.  General  Proof  of  the  Subtraction  Formulas.  — These  formulas  were 
shown  in  Art.  62  to  be  true  when  x  and  y  were  each  less  than  90°,  both  for 
x>y  and  for  x  <  y.  That  they  are  true  'for  all  values  of  the  angles  may  be 
shown  by  proving  the  special  cases  separately.  For  illustration,  let  x  be  in  the 
second,  and  y  in  the  third  quadrant.  Place  x  =  90°  +  a,  and  y  =  180°  +  /3,  so 
that  a  <  90°,  and  ^  <  90°.    Then 

sin(x-y)=sin[90°+a-(180°+/3)]=sin[-90°  +  (a-i3)]  =  -sin[90°-(a-i3)] 
=  —  cos  (a  — /3)  =  —  COS  a  cos  iS  —  sin  a  sin  0 
=  -  cos  (X  -  90°)  cos  (y  -  180°)  -  sin  (x  -  90°)  sin  (y  -  180°) 
=  —  sin  X  (  —  cos  2/)  —  (  —  cos  x)  (  —  sin  y) 

=  sin  X  cos  y  —  cos  X  sin  ?/.  q.e.d. 

Also  cos  {x-y)=  cos  [-  90°  +  (a  -  0)]  =  cos  [90°  -  (a  -  ^8)]  =  sin  (a  -  0) 

=  sin  a  cos  $  —  cos  a  sin  0 

=  sin  (x  -  90°)  cos  (y  -  180°)  -  cos  (x  -  90°)  sin  (y  -  180°) 

=  (— cos  x)  (— cos  y)  —  sin  X  (— sin  y) 

=  cos  X  cos  y  +  sin  X  sin  y.  q.e.d. 


68  PLANE   AND   ANALYTICAL  TRIGONOMETRY. 

Let  the  student  prove  the  subtraction  formulas  in  the  following  cases  ; 

1.  X  in  the  fourth,  and  y  in  the  first  quadrant. 

2.  X  in  the  fourth,  and  ?/  in  the  second  quadrant. 

3.  X  in  the  fourth,  and  y  in  the  third  quadrant. 

4.  X  in  the  third,  and  y  in  the  third  quadrant. 

5.  a;  in  the  third",  and  y  in  the  fourth  quadrant. 

6.  X  in  the  second,  and  y  in  the  fourth  quadrant. 

65.    Tangent  of  the  Sum  and  of  the  Difference  of  Two  Angles. 

sin  (x  +  ?/)      sin  x  cos  y  +  cos  x  sin  y 

tan  (a:  +  ?/)  = t^~^  = ^    —• -^• 

^        '^         cos  {x  +  y)      COS  X  COS  ^  —  sm  a:  sin  i/ 

Divide  both  numerator  and  denominator  by  cos  x  cos  y. 

sin  X  cos  ^/      cos  a:  sin  y        sin  a:      sin  y 

,  ,       cos  a:  cos  v      cos  a:  cos  y        cos  a;      cos  ?/ 

.-.  tan  (2:+?/)= ^^ ^ ^  = : -^; 

cos  X  COS  y      sin  a;  sin  «/      ^       sin  a;  sm  z/ 

cos  X  cos  ?/      cos  X  cos  y  cos  a;  cos  y 

In  the  same  way,  we  may  show  that 

.      ,  .      tan  05  -  tan  y  ^o\ 

1.   tan75°=tan(45°+30°)  =  ;^"^^°+\^"-l":=-^^^g±J^2+V3. 
^  '     1  — tan  45°  tan  SO"'  i       VS  — 1 


V3 

2.  tan  15-tan  (45°-30°)  =  J^li5;;i5a^„=  ^  =2^  =  2- V3. 

^  ^     1  +  tan  4d°  tan  30°  1        ^_|_  ^ 

V3 

3.  If  sina  =  |f  and  sinj3  =  |,  find  tan  (a  +  5)  and  tan  (a-;8),  when  a<90° 
and  B  <  90°.  Ans.  tan  (a  +  ;8)  =  -  f  | ;  tan  (a  -  )3)  =  f  |. 


66.  Geometrical  Proof.  — In  Fig.  44,  let  0^1  =  1,  AOB  =  x,  BOC  =  y. 
Draw  .5C  perpendicular  to  OJ?,  and  CD  parallel  to  OA;  :.  DBC  =  x^ 
DCE  =  x  +  y.    Then 

tan  (x  +  y)=  AE  =  AB  +  BD  +  BE. 


FUNCTIONS  OF   SEVERAL   ANGLES. 


69 


But  BD  =  BC  cos  X  =  OB  tan  y  cos  x  —  sec  x  tan  y  cos  x  —  tan  y, 

and    DE  =  CD  tan  (x  +  y)  =  BC  sin  x  tan  (x  +  y)  =  OZ?  tan  y  sin  x  tan  (x  -f  y) 
=  sec  X  tan  y  sin  x  tan  (x  +  y)  =  tan  x  tan  y  tan  (x  +  y). 
.  •.  tan  (x  -f  y)  =  tan  x  +  tan  y  +  tan  x  tan  y  tan  (x  +  y). 

tan  X  +  tan  y 


/.  tan  (x  +  y)  = 


1  —  tan  X  tan  y 


Fig.  45. 


In  Fig.  45,  let  0.4  =  1,  AOB  =  x,  COB  =  y.     Draw  50  perpendicular  to 
OB,  and  Z>(7  parallel  to  OA ;    .-.  DBC  =  x,  DCE=x-y.    Then 

tan  (.X  -  ?/)  =  ^E-  =r  ^i^  -  i>5  -  ^D. 

Rut  DB  =  BC  cos  X  —  OB  tan  y  cos  x  =  sec  x  tan  y  cos  x  =  tan  y, 

and  ^i)  —  DC  tan  (x  —  y)  =  BC  sin  x  tan  (x  —  y)  =  Oi?  tan  y  sin  x  tan  (x  —  y) 
=  sec  X  tan  y  sin  x  tan  (x  —  y)  =  tan  x  tan  y  tan  (x  —  y). 
.-.  tan  (x  —  y)  =  tan  x  —  tan  y  —  tan  x  tan  y  tan  (x  —  y). 
tan  X  —  tan  y 


tan  (x  —  y) 


1  +  tan  X  tan  y 


EXAMPLES. 

Find  by  inspection  one  value  of  x  in  Exs,  (1-6) : 

1.  sin  (w  —  l)a  cos  a  +  cos  (n  —  l)a  sin  a  =  sin  x.  ^ws.  x  =  wa. 

2.  cos  (10^=  +  a)  cos  (10°  -  a)  +  sin  (10°  + a)  sin  (10°- a)  =  cos  X.  J.>iS.  x  =  2a. 

3.  sin(a-)8  +  10°)cos(i8-a  +  10°)-cos(a-i3  +  10°)sin  ()3-a  +  10°)=  sin  x. 

Ans.  X  =  2(a  —  i8). 

4.  cos 45°  cos  (90°  -  a) -  sin 45° sin  (90°  -  a)=  cos x.     .4ns.  x  =  135°  -  o. 

5.  sin  (90°  +  h,  a)  cos  (90°  -  i  a)  +  cos  (90°+  i  a)  sin  (90°  -  i  a)  =  sin  x. 

Ans.  x=  180°. 

6.  cos(45°-a)cos(45°+a)-sin(45°-a)sin(45°+a)=cosx.  ^ns.x  =  90°. 

7.  Given  the  functions  of  30°  and  45°,  find  those  of  75°. 
Ans.  sin  75°  =  ^^+  ^ ;  cos  75°  =  ^~^  ;  tan  75°  =  ->^J-  =  2  +  V3. 


2V2 


2V2 


Vs-i 


70  PLANE   AND   ANALYTICAL   TRIGONOMETRY. 

8.  Given  the  functions  of  30°  and  45°,  find  those  of  15°. 

Ans.  sin  15°  =  ^~  ^  ;  cos  15°  =  ^^  ;  tan  15°  =  2  -  VS. 

2V^  2\/2 

9.  If  tan  a  =  I  and  sin  ;8  =  }|,  find  the  functions  oi  a  +  0  when  a  is  in 
the  third,  and  j8  in  the  second  quadrant. 

Ans,  sin  (a  +  j8)  =  -  f  | ;  cos  (a  +  j8)  =  f  f  ;  tan  (a  +  3)  =  -  f  f . 

10.  If  cos  a  =  —  If  and  sin  )8  =  -  y\-,  find  the  functions  of  o  —  /3  when  a 
is  in  the  third,  and  j8  in  the  fourth  quadrant. 

Ans.  sin  (a -)3)  =  -!§!;  cos  (a  -  )8)  =  - ||| ;  tan  (a  -  )8)  =  +  f  f  f . 

11.  If  cos  o  =  f  and  sin  ;8  =  —  |,  find  the  functions  of  o  +  ;8  and  ot  a  —  0 
when  a  is  in  the  fourth,  and  0  in  the  third  quadrant. 

Ans.  sin  (a  +  )8)  =  +  /^  ;  cos  (a  +  3)  =  -  If  ;  tan  (a  +  ;8)  =  -  /^  ; 

sin  (a  -  3)  =  +  1  ;  cos  (a  -  i8)  =  0 ;  tan  (a  -  fi)  =  oo. 

Transform  the  first  member  into  the  secpnd  (or  last)  in  Exs.  (12-32): 

12.  sin  (a  +  ;8)  sin  (a  ~  $)=  sin2  a  —  sin^  $  =  cos^  &  —  cos^  a. 

13.  cos  (a  +  ;8)  cos  (a  —  )8)  =  cos^  a  —  sin^  $  =  cos^  0  —  sin^  o. 

14.  sin  (60°  +  a)  -  sin  a  =  sin  (60°  -  o). 

15.  (r'  cos  v'  —  r  cos  vy  +  (r'  sin  v'  —  r  sin  u)2  =  r^  +  r'^  —  2  rr'  cos  (v'  —  v). 

16.  cos^  a  +  cos2  /3  —  2  cos  o  COS  i8  cos  w  =  sin2  w,  when  w  =  o  +  j8.     [Place 

a  =  w  —  iS.] 

tPf    4-  ,         tan  0  sec  a  *„„  /     ,    .\ 

17.  tan  o  H ■^-—. =  tan  (a  +  0). 

cos  a  —  tan  <p  sin  a 

18.  sin2  d  +  sin2  (u,  —  d)-\-2  sin  ^  cos  w  sin  (u)  —  6)=  sin*  w. 

19.  cos2  d  +  cos2  (w  —  ^)  _  2  cos  6  cos  a;  COS  (w  —  ^)  =  sin^  w. 

20.  tanx±tany  =  ^HL(£±l). 

cos  X  COS  y 

21.  cotx±cot2/  =  ?HLC^±^. 

sinxsmy 

22.  cotx±tany  =  ^^15i^^i^. 

sm  X  cos  y 

23.  tan  (80°  +  x)  -|-  tan  (.30°  -x)=  sin  60°  sec  (30°  +  x)  sec  (30°  -  x). 
«-     1  —  tan  a 


I  4-  tan  a 


=  tan  (45°  -  a).         [Note  that  1  =  tan  45°.] 


25.  ^  ~  ^^^  "  =  -  tan  (45°  -  a).         [Note  that  1  =  cot  45°.] 
1  +  cot  a  ^  ^  *-  ■" 

26.  sin  (60°  +  a)  -  sin  (60°  -  a)  =  sin  a. 

27.  tan  (45°  +  a)  -  tan  (45°  -  a)        4  tan  a 


1  -  tan2  a 

sin  (x  +  y)  _  tan  x  +  tan  y  _  cot  x  +  cot  y 
cos  {x  —  y)      1  +  tan  x  tan  y     1  +  cot  x  cot  y 


FUNCTIONS  OF   SEVERAL   ANGLES.  71 

29.    sin(a  +  6  +  c)=sin[(a  +  6)+c]=sin(a  +  6)cosc  +  cos(a  +  &)8inc 
=  (sin  a  cos  6 + cos  a  sin  6)  cos  c  +  (cos  a  cos  6 — sin  a  sin  6)  sin  c 
=  sin  a  cos  &  cos  c+cos  a  sin  b  cos  c+cos  a  cos  6  sin  c— sin  a  sin  6  sin  c. 

80.   cos  (a  +  6  +  c)  =  cos  a  cos  b  cos  c  —  sin  a  sin  b  cos  c  —  sin  a  cos  b  sin  c 

—  cos  a  sin  b  sin  c. 

oi     ♦««  /.,  j^  ^  L  /.A       ^^^  ^  +  t^"  ^  +  t-^-^  c  —  tan  a  tan  6  tan  c 

31.  tan(a  +  o  +  c)  = = ■ • 

1  —  tan  a  tan  b  —  tan  a  tan  c  —  tan  b  tan  c 

32.  sin  (a  +  6—  c)  =  sin  a  cos  6  cos  c  +  cos  a  sin  b  cos  c  —  cos  a  cos  b  sin  c 

.    .  4-  sin  a  sin  b  sin  c. 

33.  If  sin  a  =  f ,  sin  /3  =  -  |f ,  sin  7  =  -  f ,  find  sin  (0-/3-7),  wlien  a,  /3, 
and  7  are  in  the  second,  tliird,  and  fourth  quadrants  respectively.      Ans.  —  |f. 

34.  If  sin  a  =  f,  cos  /3  =  |,  tan  7  =  |,  find  cos  (a  -  ^  +  7),  when  a,  /3,  and  7 
are  in  the  second,  fourth,  and  third  quadrants  respectively.  Ans.  +  |. 

67.    To  express  the  Functions  of  an  Angle  in  Terms  of  those 
of  Half  the  Angle.  — In  (1)  and  (2),  Art.  61,  let  1/  =  x. 

••.  sin  2  0?  =  sin  x  cos  x  +  cos  x  sin  a;  =  2  sin  a?  cos  x,  (1) 

co^  2  05  =  cos^  X  -  sm^  x,  (2) 

cos  2  a?  =  1  —  sin^  x  —  sin^  a:  =  1  -  2  sin'^  a:,  (3) 

or  cos  2  a?  =  cos^  2;  —  1  +  cos^  x  =  2  cos^  x-1,  (4) 

From  (1),  Art.  65, 

tflng^-   tan  a: -t- tan  a:    ^    2tanag   ^ 
1  —  tan  x  tan  x      1  -  tan*  x 

1.  To  express  the  functions  of  40*^  in  terms  of  those  of  20°,  we  have 

sin  40°  =  2  sin  20°  cos  20°  ; 
cos  40°  =  cos2  20°  -  sin2  20°  ; 

tan40°=^iH1^21_. 
1  -  tan2  20° 

2 

2.  sin  e  =  — -,  d  being  in  the  second  quadrant.    Find  sin  2  ^,  cos  2  0,  tan  2  B. 

V6 

^ns.  sin  2  ^  =  -  4  ;  cos  2  ^  =  —  | ;  tan  2  ^  =  +  |. 

3.  tan  ^  =  +  2,  ^  being  in  the  third  quadrant.     Find  sin  2  6,  cos  2  ^,  tan  2  ^. 

Ans.  sin  2  ^  =  +  I ;  cos  2  0  =  -  |  j  tan  2  ^  =  -  f 

4.  cot  ^  =  —  I,  ^  being  in  the  fourth  qundrant.     Find  sin  2  5,  cos  2  0,  tan  2  0. 

A  ns.  sin  2  e  =  -  ^1 ;  cos  2  ^  =  -  ,'^  ;  tan  2  ^  =  +  V". 


^2 


PLANE   AND   ANALYTICAL   TRIGONOMETRY. 


Note.  — To  find  Ihe  area  of  a  right  triangle,  given  c  and  a,  we  have 


area  =  \  ah. 
But  a  =  c  sin  a,  and  h  =  c  cos  a. 
. '.  area  =  \  c-  sin  a  cos  a  ; 
. '.  area  =  |  c^  sin  2  a. 


(6) 
(7) 


In  (6)  we  should  have  to  find  both  sin  o 
and  cos  a  from  the  tables,  in  (7)  we  find 
only  sin  2  a,  so  that  time  is  saved  by  using 
(7)  instead  of  (0). 

68.   Geometrical  Proof.  —  Let  the  radius  OA  of  the  circle  be  unity. 

sin  2x  =  BP=2CD  =  20D  sin  x 

=  2  OA  cos  X  sin  x. 

cos2 X  =  OB  =  OC  -  BC  =  OC-CA 
=  OD  cos  X  —  ^D  sin  a; 
=  0.1  cos2  x-OA  sin2  x 
=  0A  {cos^x-  sin2a:). 

5P^     2CD     ^     2  QC  tan  a; 
OB     OC-CA     OC- CD  t&nx 
2  OC  tan  a;  2  tan  ar 


tan  2  X 


OC- octant  a;      l-tan2a; 

69.    To  express  the  Functions  of  an  Angle  in  Terms  of  those 
of  Double  the  Angle.  —  From  Art.  67, 

cos  ^a:  =  1  —  2  sin^rr  ; 
•••  2  sin- a?  =  1  -  cos  2  i».  (1) 

cos  2x  =  2  cos^ X  —  1 ; 
2  cos^  a?  =  1  +  cos  2  x,  (2) 

.2^  —  1  -  COS  2  X 


Also 

a 

From  (1)  and  (2), 


tan  2:  =  ±-v/- 


1  +  cos  2  X- 

'1  —  cos  2x 


H-  cos  2  a; 


.  tan  X  = 
•.  tan  X  - 


^^ 


—  cos  2x     1  4-  cos 


+  cos  2x     1  +  cos  '^x 
sin  2  a? 


l!=>a 


—  cos2  2a: 


(1  +  cos  22:) 


2' 


1  +  cos  2  a5 


Also 


_    /l— cos  2  2:     1  —  cos  22:_    /(I  —  cos  22:)2_ 
^  1  +  cos  2x     1  —  cos  2x      ^    1  —  cos^  2  x 


.*•  tan  0? 


1  -  cos  2  05 
• 

sin  2  a? 


(8) 
(4) 

(5) 

(fi) 


FUNCTIONS   OF   SEVERAL   ANGLES. 


73 


Note. — The  double  sign  is  not  used  in  (5)  and  (6),  for 
sin  2  X        2  sin  x  cos  x 


and 


1  +  cos  2  a;         2  cos-'  x 
1  —  cos  2  X         2  sin^  x 


sin  2  X         2  sin  x  cos  x 


=  tan  X, 
r=  tan  X. 


1.    To  express  the  functions  of  20°  in  terms  of  those  of  40°,  we  have 
2sin2  20°  =  1  -cos  40°; 
2cos2  20°  =  1  +  cos  40°; 
1  -cos  40° 


tan2  20^ 
tan  20°  = 


1  +  cos  40° 

sin  40°     _  1  -  cos  40^ 
1  +  cos40°~ 


sin  40= 


2.    tan  2  ^  =  —  2,  2  ^  being  in  the  second  quadrant.    Find  the  functions  of  B. 

1  o 

.-.  cos  2  ^  = —,  and  sin  2  ^  =  -f  -=-. 

cos  ^  =  ±  Y"  ( 1  -  -^) ,  from  (2)  ; 


.*.  sin 


tan  5  = 


2 

V6 


1  _  J_     Vs  -  1 

V5 


from  (5). 


Since  2  ^  is  in  the  second  quadrant,  B  may  be  either  in  the  first  or  in  the 
third  quadrant ;  hence  sin  6  and  cos  6  have  the  double  sign,  and  tan  B  is  positive. 


3.    Given  the  functions  of  30°,  find  those  of  15°. 


Ans.  sin  15°^  ^-  ^ 


cos  15= 


^^ll±l-  tan  15°  =  2 -V3. 


2V2    ■  2V2 

4.   Given  the  functions  of  45°,  find  those  of  22J°. 

Ans.  sin22i°  =  ^V2-V2;  cos22|°  =i  V2  +  \/2  ;  tan22|°  =  V2  -  1. 
70.    Geometrical  Proof.  —  Let  the  radius  CA  of  the  circle  be  unity. 

e^ ^P_  ^~6B'Ba _ABa ^Jba 


OP     VOBToA      ^  OA 

= Jg-^  -  CB  ^    j\  -cos2x 


OB 


OB 


OP     y/OB.OA      ^0.\ 


=M-r-^ 


lOC+CB     J\  +cos2x 
=  \— ^ =\ 1 


74  PLANE   AND   ANALYTICAL   TRIGONOMETRY. 

BP  BP  sin  2  a; 


tana;  = 
tana: 


OB      0C+  CB      1  +  cos2a: 

BA ^  CA-  CB ^  1  -cos2x 
BP  BP  sin2x 


71.  Multiple   Angles.  —  Suppose   that  we  wish   to  express 
sin  3  X  in  terms  of  powers  of  sin  x. 

sin  3  a:  =  sin  (2  a;  +  ^)  =  sin  2  x  cos  x  +  cos  2  a:  sin  a: 

=  2  sin  X  cos^  a:  +  (1  —  2  sin^  x}  sin  x 
C^'^'Z  ^  /  j^  ^OyUji^J     ^^    =  2  sin  a;  —  2  sin3  a:  +  sin  a;  —  2  sin^  a: 

^  =  3«ina;  —  4sin3,a^.  Q.E.I. 

1.  Show  that  cos  3  x  =  4  cos^  x  —  3  cos  x. 

2.  Show  that  tan3x=^^^^^-^^"'^. 

1  -3tan2a; 

3.  Show  that  sin  4  x  =  4  sin  x  cos  a;  —  8  sin^  x  cos  aj.     [Use  4x  =  i2x  +  2x.] 

4.  Show  that  cos  4  x  =  1  —  8  sin^  x  +  8  sin*  x. 

E     ou       *u  *  *      A         4tanx(l  -  tan2x) 

6.   Show  that  tan 4 x  = -.^  \ — ^ • 

1—6  tan2  X  +  tan*  x 

6.  Show  that  sin  6  x  =  5  sin  x  —  20  sin^  x  +  16  sin^  x.     [Use  5  x  =  3  x  +  2  x.] 

7.  Show  that  cos  5  x  =  5  cos  x  —  20  cos^  x  +  16  cos^  x. 

8.  Find  the  functions  of  18°,  of  36°,  and  of  72°. 
Place  X  =  18°  ;  then,  since  cos  54°  =  sin  36°,  we  have 

cos  3  X  =  sin  2  x. 

.•.  4  cos-5  X  —  3  cos  X  =  2  sin  X  cos  x. 

.'.  cos  X  (4  cos2  X  —  3  —  2  sin  x)  =  0. 

/.  1—4  sin2  X  —  2  sin  X  =  0. 

.-.  sinx  =  ^(—  1  ±  Vo). 

.-.  sin  18°  =  cos72°  =  K>/5  -  1)  ;  cos  18°  =  sin  72°  =  jVlO  +  2V5. 

Hence  sin  36°  =  2  sin  18°  cos  18°  =  ^VlO-2V5  ; 

cos36°  =  1 -2sin218°  =  i(^^+ 1). 

72.  To  change  the  Product  of  Functions  of  Angles  into  the 
Sum  of  Functions.  —  From  Arts.  61  and  62, 

sin  (a;  +  ^)  =  sin  x  cos  y  +  cos  x  sin  i/;  /^ 

sin  (x  —  y^=  sin  x  cos  y  —  cos  x  sin  y.  ^^^ 

.'.  sin  (a;  H- «/)  +  sin  (a;  —  ?/)  =  2  sin  a:  cos  «/,  (1) 

and  sin  (x  -\-  y}—  sin  (a;  —  «/)  =  2  cos  x  sin  y.  (2) 


FUNCTIONS  OF  SEVERAL  ANGLES.  75 

Also  cos  (a:  +  y)  =  cos  x  cos  y  —  sin  a:  sin  y\   ^ — 

cos  (x  —  y)=  cos  X  cos  y  +  sin  x  sin  y.  -^ 
.*.  cos  (a;  4- y )  +  cos  (a;  —  ?/)  =  2  cos  a;  cos  y,  (3) 

and  cos  (x  +  y)—  cos  (a:  —  ?/)=  —  2  sin  a;  sin  y .  (4) 

Reversing  (1),  (2),  (3),  and  (4),  we  have 

sina3COS2/  =  ^sin(a!;  +  2/)+^sitt(a5-2/),     \  (5) 

cos ic  sin  2/  =  I  sin  (05  + 2/)-^  sin  (05-1/).      [  (6) 

cosxcosy  =  |cos(a5  + 2/)+ |cos(a5 -y).  (7) 

sina5sin2/  =  -|cos(a5  +  i/) +|cos(a5- J/).  (8) 

In  applying  these  formulas,  let  x  represent  the  larger  angle. 

1.  sin  4  ^  cos  2  0  =  ^  sin  (4  ^  4-  2  ^)  +  I  sin  (4  ^  -  2  ^),  from  (5), 

=  1  sin  6  ^  +  1  sin  2  d. 

2.  cos  6  d  sin  2  ^  =  ^  sin  (6  ^  +  2  ^)  -  ^  sin  (6  ^  -  2  6),  from  (6), 

=  ^  sin  8  ^  -  ^  sin  4  ^. 

3.  cos8^  cos2^  =  ^coslO^  +  ^cos6^,  from  (7). 

4.  sin  6  ^  sin  4  ^  =  -  I  cos  10  ^  +  ^  cos  2  6,  from  (8). 

5.  cos  2  0  sin  4  0  =  i  sin  6  ^  -  i  sin  (  -  2  ^),  from  (6), 

=  i  sin  6  ^  +  ^  sin  2  <?,  as  in  Ex.  1. 

6.  sin2^cos6^=  isin8^  +  |sin  (- 4^),  from  (5), 

=  i  sin  8  0  —  I  sin  4  ^,  as  in  Ex.  2. 

7.  cos2^  cos  8^=  1208  10^+ I  cos  (- 6^),  from  (7), 

=  ^  cos  10  ^  +  I  cos  6  ^,  as  in  Ex.  3. 

8.  sin  4  e  sin  6  ^  =  -  ^  cos  10  ^  +  ^  cos  ( -  2  ^),  from  (8), 

=  —  ^  cos  10  ^  +  I  cos  2  d,  as  in  Ex.  4. 

9.  sin2  ^  cos  ^  =  sin  d  [sin  6  cos  d]  =  sin  d  [^  sin  {6  -{-  6)+  \  sin  (^  —  ^)] 

=  sin  ^  [^  sin  2  ^  +  I  sin  0°]  =  |  sin  0  sin  2  d 

=  K  -  i  cos  3  ^  +  ^  cos  ^]  =  -  I  cos  3  ^  +  i  cos  ^. 

10.  Reduce  sin^  a  cos  a  to  ^  sin  2  a  —  ^  sin  4  a. 

sin3  a  cos  o  =  sin2  a  •  sin  a  cos  a  ; 
using  (8)  and  (5),  or  the  relations  in  Arts.  69  and  67,  we  have 

sin3  a  cos  o  =  I  (1  —  cos  2  a)  .  ^  sin  2  o  =  ^  sin  2  a  —  ;^  sin  2  a  COS  2  a 
=  ^  sin  2  a  —  ^  sin  4  o. 

11.  Reduce  sin^  d  cos2  ^  to  ^  (1  —  cos  4  d). 

12.  Reduce  sin2  d  cos^  ^  to  |  (cos  ^  -  ^  cos  3  ^  —  ^  cos  5  ^). 

13.  Reduce  sin^  0  cos^  6  to  ^^5  (3  sin  2  ^  -  sin  6  ^). 

14.  Reduce  cos^  d  to  ^-^  (10  cos  ^  +  5  cos  3  ^  +  cos  5  d). 

15.  Reduce  cos^  6  to  \  (cos  3  ^  +  3  cos  0). 

16.  Reduce  sin^  0  cos^  0  to  -^^  (3  sin  2  ^  -  sin  4  ^  -  sin  6  ^  +  ^  sin  8  6). 


76 


PLANE   AND    ANALYTICAL   TRIGONOMETRY. 


73.   To  change  the  Algebraic  Sum  of  Functions  of  Angles  into 
the  Product  of  Functions.  —  Let  x  +  y  =  u  and  (2;  —  ^)  =  v, 

.'.  X  =  l(^u  -{-  v~)  and  y  =  ^(u  —  v). 
Substituting  in  (1),  (2),  (3),  and  (4),  Art.  72,  we  have 

sinw  + sinv  =  2siii|(w  +  v)cos^(m- v).  (1) 

sin  1*  -  sin  V  =  2  cos|  {u  +  v)  sin  |  (w  -  v),  (2) 

COSW  +  cosv  =  2cos^(w  +  i^)cos|(w  -V).  (3) 


cosM  -  cosv  =  -  2sin|(w  +  v)sin|(M  —  v). 


In  applying  the  formulas,  let  u  represent  the  larger  angle. 

1.  Reduce  sin  3  ^  +  sin  ^  to  2  sin  2  ^  cos  6. 

Let  u  =  ZO  and  v  =  ^  in  (1). 

2.  Reduce  cos  ^  —  cos  3  ^  to  4  sin^  d  cos  d. 

cos  d  —  cos  3  ^  =  —  (cos  •')  0  —  cos  d).     Let  ?(  =  3  ^  and  v  =  ^  in  (4). 
.  •.    -  (cos  3  ^  -  cos  d)  =  -  (  -  2  sin  2  ^  sin  ^)  =  +  2  sin  2  ^  sin  0 
=  4  sin  ^  cos  ^  sin  <^  =  4  sin^  d  cos  d. 

3.  Reduce  sin  3  ^  +  cos  ^  to  a  product. 

sin  3  ^  +  cos  e  =  sin  3  ^  +  sin  (90°  -6)  =2  sin  (45°  +  d)  cos  {2  6  -  45°) 
=  2  sin  (45°  +  6)  cos  (45°  -26). 


74.    Geometrical  Proof. — In  the  figure,  OD  bisects  the  angle  QOP,  and 
is  therefore  perpendicular  to  QF.     Using  the  notation  there  shown,  we  have 

QOP=u-v\  .:  qOD^DOP=\{u-v); 
AOD  =  AOq+  qOD  =  v+\  {u-v)  =  \{u^v); 

FPQ=GDQ  =  AOD=]  (u+v).    Then,  if  the 
radius  =  1, 

sin  w+sin  v  =  BP+  CQ  =  2  ED=2  OD  sin  AOD 
=  2  OP  cos  DOP  sin  AOD 

=  2sm  \{u-\-v)cosl(u  —  v).        (1) 

sin  u-sinv=BP-  CQ  =  2  GD=2  DQ cos  GDQ 
=  2  OQ  sin  QOD  cos  GDQ 

=  2  cos  I  (u-^v)  sin  |  (u—v).      (2) 

cos  M  +  cos  r=OZ?+OC=  2  0E=2  OD  cos  A0D=2  OP  cos  DOP  cos  AOD 

=  2  cos  I  (m  +  u)  cos  I  (w  — r).  (3) 

cosM-cosv=  OS- 0C= -2  GQ=-2DQs\n  GDQ=-2  OQ  sin  QOD  sin  GDQ 

=  —  2  sin  I  (u  +  v)  sin  |  (u  —  v) .  (4) 


Fig.  49. 


FUNCTIONS  OF   SEVERAL   ANGLES. 


7T 


EXAMPLES. 

Show  that  the  first  member  of  the  equation  may  be  reduced  to  the  second 
(or  last)  in  Exs.  (1-7): 

1.   sin  (45°  +  a-)  +  sin  (45°  —  x)  =  2  sin  45°  cos  x  =  V2  cos  x. 
3.   sin  (90°  +  X)  -  sin  (180°  +  re)  =  2  cos_(135°  +  a;)  sin  (  -  46°) 

=  -\/2cos(135°  +  a;). 

3.  cos  (180°  +  x)  +  cos  (180°  -  a-)  =  2  cos  180°  cos  x  =  -  2  cos  x. 

4.  cos  (270°  +  X)  -  cos  (270°  -  x)  =  -  2  sin  270°  sin  x  =  +  2  sin  x. 
6.   sin  3  X  +  2  sin  5  X  +  sin  7  x  =  4  sin  5  x  cos^  x. 

6.  cos  3  X  +  2  cos  5  X  +  cos  7  X  =  4  cos  5  x  cos^  x. 

7.  cos  (6  —  c)  —  cos  a  =3  +  2  sin  i  (a  +  &  —  c)  sin  |(a  —  6  +  c) . 

8.  Show  tliat  sin  (\"—  X')  —  sin  (X"—  X)  +  sin  (X'—  X)  may  be  reduced  to 
4  sin  \  (X'  — X)  sin  \  (\"  —  \')  sin  ^  (X"— X).  [The  formula  sin  x=2  sin  |  x  cos  |  x 
is  iised  in  the  process.] 

If  a 4-j8  +  7  =  180°,  reduce  the  first  member  to  the  second  in  Exs.  (9-14): 

9.  sin  o  +  sin  /3  +  sin  7  =  4  sin  J  (a  +  j8)  cos  J  a  cos  ^  0. 

7  =  180°  -(a  +  B);    .:  sin  y  =  sin  (a  +  $).     Then 
sino+sin  /3  +  sin  (a4-)8)=2sin^  (a  +  i3)  cos  J  (a  — )3)4-2  sin  J  (a  +  )8)  cos^(a  +  ;8) 
=  2sin^(a  +  i8)[cos^(a  -  /3)+cos  J(a+  3)] 
=  2  sin  ^  (a  +  )8)  (2  cos  J  a  cos  J  i3). 

10.  cos  a  +  cos  )8  +  cos  7  =  4  cos  ^  (a  +  j8)  sin  J  a  sin  ^  )8  +  1.     [Note   that 
cos  7  =  —  COC  (a  +  )8)  =  —  2  cos2  ^  (a  +  i8)  +  1.] 

11.  cos  2  a  +  cos  2  )8  +  cos  2  7  =  —  4  cos  a  cos  B  cos  7  —  1. 

12.  sin  2  a  +  sin  2  j8  +  sin  27  =  4  sin  a  sin  $  sin  7. 

13.  2  sin2  a  +  2  sin^  3  +  2  sin^  7  =  4  +  4  cos  a  cos  $  cos  7. 

14.  sin  3  a  +  sin  3  )3  +  sin  3  7  =  —  4  cos  |  a  cos  |  ^3  cos  1 7. 

15.  If  a  +  3  +  7  =  360°,  sin  a  +  sin  ^  +  sin  7  =  4  sin  J  a  sin  J  /3  sin  J  7. 

16.  If  a  +  ;8  +  7  =  360°,  sina  +  sin)3  +  2  sin -^-7=4  sin  |(a  +  )8)  cos2  1  (a-/3). 

75.  Circular,  or  Inverse  Trigonometric,  Functions.  —  If  y 
is  the  sine  of  the  angle  or  arc  x,  then  x  is  the  arc  whose 
sine  is  y.  This  is  written  x  =  sin~^  y,  read 
"  X  is  the  arc  whose  sine  is  y."  So  also  if 
tan  X  =  m,  then  '•'■  x  is  the  arc  whose  tangent  is 
w,"  written  x  =  tan~^  m. 

In  consequence  of  this  notation,  if  we  have 

-: and  wish  to  bring  sin  x  into  the  numer- 

sm  a;  o  . 

ator,  we  must  write  it  in  a  parenthesis  with  the  exponent  —  1 ; 


Fig.  50. 


78  PLANE   AND  ANALYTICAL   TRIGONOMETRY. 

-  =  (smx)~^.     All  other  exponents  may  be  written  above 

sin  a;  -. 

the  name  of  the  functions  ;     .  .,    =  sin"^  x  =  (sin  x)~^. 

sin'*  X 

1.   y  =  tan~i  m  +  tan-i  n.     Find  tan  y. 

Let       tan-i  m  =  a  and  tan-i  n  =  b  ;  .-.  tan  a  =  m,  tan  b  =  n. 

tan  a  +  tan  b        m  -{■  n 


.-.  y  =  a  +  b;  .-.  tan  1/ 


1  —  tan  a  tan  6      1  —  win 


2.   tan-i  —  =  tan-i  — \-  tan"!  x.     Find  x. 

w  w  4-  w 

.  •.  tan~i  X  =  tan~i tan~i Let  a  =  tan-i  — ,  6  =  tan-i  — - — 

m               wi  +  n                             m  m  -\-  n 

1  1 

/              ,                  *      /        1.N       tan  a  —  tan  6          m  m  +  n 
.*.  tan-i  x  =  a  —  6;  .-.  x  =  tan  (a  —  6)  = 


1  +  tan  a  tan  6      ,    , 


•.   X 


m{m  +  n) 


m'^  +  ?»n  4-  1 
S.*  y  =  sin-i  ^  +  tan-i  |.     Find  sin  y.  Aris.  sin  y  =  J^  (4  +  3V3). 

4.    tan-i  —  =  tan-i  — ^ tan"!  x.      Find  x=  ^ 


m  m  —  n  m'^  —  mn  + 1 

6.    tan-i  a  =  tan-i  \  +  tan-i  ^i-     Find  a  =  \. 

6.*  y  =  sin~i  m  +  sin~i  n.     Find  sin  ?/  =  m  Vl  —  n'-^  +  n  Vl  —  m^. 

7.*  y  =  cos-i  771  +  cos~i  n.     Find  sin  y  =  nVl  —  w'^  -f  my/ 1  —  n^. 

8.*  y  =  cos~i  wi  —  sin"i  n.     Find  cos  y  =  wVl  —  n'^  4-  ?iVl  —  rri^. 
'  9.  tan-i  a  =  tan~i  J  —  tan-i  ^.     Find  a  =  ^ 
10.*  m  =  tan-i  J  +  tan-i  i      Find  m  =  45^. 
11.*  m  =  tan-i  J  +  tan-i  |  +  tan-i  y^-     ^ii^d  m  =  45°. 
12.*  w  =  2  tan-i  ^  -  tan-i  f     Find  m  =  45°. 
Let  tan-i  ^  =  a,  tan-i  4^  =  6;    .-.  m  =  2a  +  6;    .-.  tan  wi  = ,  etc. 
13.*  m  =  2  tan-i  |  +  tan-i  j.     Find  m  =  45°. 
14.*  Show  that  tan-i  |(1  -  w)  =  sec- 1 1  Vs  -  2  m  +  wi'-^. 
Let  tan-i  l(\  —  m)  =x  ;     .  •.  tan  x  =  ^  (1  —  w) ;  sec  x=  Vl  +tan2  x ; 


. •.  sec X  =  IV 5  —  2m-\-  m'^.     . •.  x  =  sec-i  I  Vo  —  2m  -\-  m^. 

15.  Show  that  tan-i  m  =  i  tan-J    ^^    . 

'  1  -  Wi2 

Let  X  =  tan-i  ,jj^  or  m  =  tan  x.     If  the  equation  is  true,  we  must  have 

X  =  i  tan-i  -1^,  or  2  X  =  tan-i    ^^^"^   ,  or  tan  2  x  =  Al^^l^, 
1  -m'^  1  -  tan2  x  1  -  tan2  x 

a  formula  proved  in  Art.  67. 

16.  Show  that  cos-i  m  =  ^  cos-i  (2  m^  —  1). 

17.*  Show  that  sin-i  ^V^  ^  tan- 1  ?^^. 
a  +  &  a  —  6 

*  When  the  angles  are  less  than  90°. 


FUNCTIONS  OF   SEVERAL  ANGLES.  79 

18.   Showthatsinf--2tan-iA/l-^^  =  a;. 

19.*  Show  that  |  vers  ^  a^  —  sin-^  i  a  is  constant  for  all  possible  values 


of  a. 


Let  6  =\  vers-i  \a^  —  sin-i  \  a,  and  let  m  =  vers-^  ^0:^^71  =  sin-^  ^  a. 

cos  2  ^=cos  m  cos  2  n+sin  wi  sin  2  n. 
But 


i;     .'.  2^=m-2ni 
sin  n  =  ^  a  ;     .  ••  cos  n  =  J  \/4  —  a^ 


.  •.  sin  2  71  =  I  a  V4  —  a^  ;  cos  2  w  =  i  (2  —  a^). 


Also 


cos  m  =  1  —  vers  wi  =  1  —  |  a^ ;     .  •.  sin  m  =  -  V4  —  a*. 


•.  cos  2^  = 


2  -  a2     2  -  a2  .  a 


2  2  2 


2^  =  0°,  or  d  =  (f 


20.*  Show  that  taii-i ^4-sin-i  ^  jg  constant  for  all  possible  values  of  a. 


a 


21.*  Show    that  vers-i  a  —  2  cot-^A/    ~  ^  is    constant    for    all    possible 
values  of  a. 

X 


a 


22.*  Show  that  vers-i  —  —  2  sin-i-v/—  is  constant  for  all  possible  values 
12  >'24  ^ 


of  z. 

76.  To  prove  that  tan  a? > £c> sin  «  when  a; <  — ,  a?  being  ex- 

2 

pressed  in  Circular  Measure.  —  Let  AOB  =  BOC  =  x^  the  radius 
being  unity.     Evidently  AT>  SB,  or  tan  x  >  sin  x. 

Also,  since  the  shortest  distance 
from  a  point  to  a  line  is  perpen- 
dicular to  the  line,  SB  <  AB,  or 
sin  x<x. 

The  arc  AC  may  be  considered 
as  composed  of  an  infinite  number 
of  infinitesimal  straight  lines  ;  hence 
AT-^  TO>&vcABO,  since  ABO  is 

a  convex  polygon  lying  in  the  tri-      At^ 

angle  formed  by  a  chord  A  0  with  fig.  6i. 

the  tangent  lines  TA  and  TO.     Then 

2  AT  >  arc  ABO,  or  AT>a>TGAB,  or  tanic>a;. 

Hence  tan  x>x,  and  x  >  sin  x.  q.e.d. 

77.  To  prove  that  sin  a?,  tan  a?,  and  a?  approach  Equality  as 
the  Angle  a?  approaches  Zero.  —  As  the  angle  AOT  decreases, 


When  the  angles  are  less  than  90*^ 


80  PLANE   AXD   ANALYTICAL   TRIGONOMETRY. 

the    points    B  and   T  approach  A^  and  hence   approach  each 
other.     But 

SB       sin  X 


AT     i'Awx 


cos  a;. 


When  the  angle  x  approaches  zero  as  its  limit,  cos  x  approaches 

unity  as  its  limit.     Hence  -j-^,  or  ,  approaches  unity  as  its 

limit,  or  sin  x  and  tan  x  approach  equality. 

The  arc  x  is  intermediate  in  value  between  sin  x  and  tan  x ; 
hence  the  three  quantities  approach  equality  as  the  angle  be- 
comes smaller.     That  is,  the  three  ratios 

sin  X     sin  x     tan  x 


tan  XX  X 

approach  unity  as  the  angle  approaches  zero. 

Hence  we  may  say  that  when  the  angle  is  small,  its  sine  and 
its  tangent  are  equal  to  the  arc  itself,  and  its  cosine  is  equal 
to  unity.  The  smaller  the  angle,  the  more  nearly  correct  will 
be  the  assumption. 

78.  Development  of  sin  a?,  of  cos  «,  and  of  tan  a?.  —  Let  us 
assume  that 

sin  X  =  a  +  hx  -\-  cx^  -\-  ds?  +  es^  +f^  +  •••  (1) 

is  true  for  all  values  of  x.  Then  it  is  true  when  x  has  the 
values  4-  y  ^^^  ~  V  j  hence 

sin^  =  a-\-hy^-cf'-\-  dy^  +  ey^  -{-fy^  +  •••  (2) 

and         sm(—  y}=a  —  by  -{-  cy^  —  dy^  +  ey^  —fy^  -\ (3) 

But  sin  y  =  —  sin  (  —  ^),  or  sin  ?/  +  sin  (—?/)=  0.  Adding 
(2)  and  (3), 

2  a  +  2  £?/  +  2  e/  +  ...  =  0.  (4) 

But  (4)  is  true  for  all  values  of  y,  since  (1)  is  true  for  all 
values  of  x.  In  order  that  all  values  of  y  may  reduce  the  left 
member  of  (4)  to  zero,  we  must  have  a  =  0,  c  =  0,  e  =  0,  ..-• 
Hence  (1)  becomes 

sin  x  =  hx  -\-  dx^  -\-fx^  +  •••  (5) 

or  ?HL£  =  6  +  ^a;2+A*+-  (6) 

X 


FUNCTIONS   Or    SEVERAL   ANGLES.  81 

But  as  X  approaches  zero,  ^^^  approaches  unity,  and  h  +  dx^ 

X 

-f/r*+  ...  approaches  h.      Hence 

1  =  J,  (7) 

and  (5)  becomes 

sin  £c  =  a?  +  doc^  +  fx^  +  •••  (8) 

Again,  let 

cos  a:  =  ^  4-  J5a:  +  Cx^  ^  Bt? -\-  Ex"^  +  Fx^  +  -.•  (9) 

Since  cos  x  =  cos  (  —  x')^  we  have 

A  +  Bx  +  Cx'^  +  Dj^  +  E3^  +  F3^-[-  ... 
=  A-Bx-\-Cx'^-  B^  H-  J^;^;*  -  i<:r5  ^  ...  (IQ) 

or  2j52:  +  2i>ar^  +  2jP2:5+  ...  =  0.  (11) 

In  order  that  this  may  be  true  for  all  values  of  x^  we  must  have 
^  =  0,  i>  =  0,  ^=0  ...,   and  (9)  becomes 

cos  a:  =  ^  +  Cx^  +  Ed"  +  -  (12) 

But  when  a:  =  0,  (12)  reduces  to 

1  =  A,  (IS) 

and  hence  (12)  becomes 

cos  a^  =  1  +  Cic-  +  Bqc^  +  ...  (14) 

Substituting  from  (14)  and  (8)  in  the  formula 
cos  2  a:  =  cos^  a:  —  sin^  a:, 
we  have      1  +  4  Ca:2  _j_  iq  ^^4  _^  ...  ^  1  ^(9  (7_  i)^2 

+  (2  ^  +  6^2  _  2  d):»^  4-  •••  (15) 

Equating  the  coefficients  of  like  powers  of  x^ 

4(7=2(7-1,  or  2(7+1=0.        (16) 

lG^=2^+(72_2^,     or     WE-C^^ld^^.       (17) 

Substituting  from  (14)  and  (8)  in  the  formula 
sin  2  a:  =  2  sin  x  cos  a:, 
we  have   2  a:  +  8  cZa;3  4.  32  /.^;5  +  ...  ^  2  a:  +  2  ((7+  d^T? 

+  2(iE  +  Cd+f^7^+...  (18) 

CROCK.    TRIG.  6 


82  PLANE  AND  ANALYTICAL   TRIGONOMETRY. 

Equating  the  coefficients  of  like  powers  of  a;, 

4.d  =  C-\-d\              or                3c^-(7=0.  (19) 

lQf=E  +  Cd+f,     or     15f-U-Cd  =  0.  (20) 

From  (16),                 0=-^.  (21) 

From  (19),                 ^  =  -1=-^.  (22) 

From  (17),               U=+^  =  +^  (23) 

From  (20),                /  =  +^=+^.  (24) 

These  values,  substituted  in  (8)  and  (14),  give 

^inx  =  a^-^  +  ^^.-^  ij^-^  (25) 


i_n 


cosaj=  1 


g?^  .  ag^ 


L2.    LI 
Dividing  (25)  by  (26), 

tan  i»  =  05  +  4  ^3  ^  ^  ar;5  _j.  ... 


15 


(26) 


(27) 


In  (25),  (26),  and  (27),  which  are  the  required  develop- 
ments, X  must  be'  expressed  in  circular  measure. 


79.  Computation  of  the  Trigonometric  Functions  (First 
Method).  —  The  functions  may  be  computed  by  (25),  (26),  and 
(27),  Art.  78.  Thus,  to  find  sin 20°,  we  place  x=\'ir,  the 
circular  measure  of  20°. 


log  7r8=  1.49145 
col93  =  7.13727 -10 
col6  =  9.22185 -10 

log  7r6  =  2.4857 
col96  =  5.2288 -10 
C0II2O  =  7.9208  -  10 
log*^  -  5.6353  -  10 

li 

.-.^=0.0000432       . 

X 

:  sin 

=  -  =  0.34906  59 
9 

^.=  0.00708  88 
|3       

0.34197  71 

log  ^  =  7.85057  -  10 
.-.  ^  =  0.0070888 

^=0.00004  32 
20°  =  0.34202  03 

li 


In  the  tables,  sin  20°  =  0.34202. 


FUNCTIONS  OF  SEVERAL   ANGLES.  83 

80.    Computation    of    the    Trigonometric    Functions    (Second 
Method).  —  From  (25),  Art.  78,  it  may  be  shown  that 

sin  1"  =  0.00000  48481  36811  07637, 
while  arc  1''  =  0.00000  48481  36811  09536. 

.  •.  arc  1''  -  sin  1''  =  0.00000  00000  00000  02. 
Again,        sin  V  =  0. 00029  08882  04563  42460, 
while  arc  1'  =  0.00029  08882  08665  72160. 

.-.  arc  r- sin  1^  =  0.00000  00000  04. 
Again,        sin  1°  =  0.01745  24064  37283  51282, 
while  arc  1°  =  0.01745  32925  19943  29577. 

.-.  arc  r- sin  1°  =  0.00000  09. 
'  Also,  from  (26),  Art.  78, 

cos  V  =  0.99999  99999  88  =  1  -  0.00000  00000  12. 
cos  1'  =  0.99999  99576  92  =  1  -  0.00000  00423  08. 
cos  1°  =  0.99984  76952      =  1  -  0.00015. 
In  computing  a  set  of  five-place  tables,  we  may  assume 
sin  1'  =  arc  V  =  0.00029  08882  with  an  error  of  5  x  lO-^^, 
and      cos  1'  =  1  with  an  error  of  4  x  10"^. 

Then  sin  2^  =  2  sin  1'  cos  1'  ;  cos  2'  =  cos^  1^  —  sin^  l^ 
sin  3'  =  sin  2'  cos  1'  +  cos  2'  sin  1'  ; 
cos  3'  =  cos  2'  cos  1'  —  sin  2'  sin  V. 
sin  4'  =  sin  (3'  +  1^ ;  cos  4'  =  cos  (3'  +  1^, 
or        sin  4'  =  2  sin  2'  cos  2' ;  cos  4'  =  cos2  2'  -  sin2  2'. 
And  so  on. 

This  method  would  be  employed  until  the  functions  of  all 
angles  less  than  30°  had  been  computed.     Then,  since 

sin  (30°  +  a:)  =  cos  a;  -  sin  (30°  -  rr), 
and  cos  (30°  -h  x)=  cos  (30°  —  x)—  sin  x, 

the  functions  of  angles  between  30°  and  45°  would  be  found  by 
combining  the  functions  already  found.  Thus,  if  2;  =  10°,  we 
have  sin  40°  =  cos  10°  -  sin  20°, 

and  cos  40°  =  cos  20°  -  sin  10°. 


84  PLANE   AND  ANALYTICAL   TRIGONOMETRY. 

It  is  possible  to  compute  independently  the  sine  and  cosine 
of  3°,  6°,  9°,  ...,  39°,  42°,  45°.  We  have  found  in  this  chapter* 
the  sine  and  cosine  of  15°,  of  18°,  and  of  36°,  and  we  have 

3°  =  18°-15°,     6°  =  3G°-30°,     9°  =  45°~36°,  12°  =  30°-18°, 
21°  =  36°-15°,  24°  =  45°-21°,  2T°  =  45°-18°, 
33°  =  18° +  15°,  39°  =  45°-   6°,  42°  =  45°-   3°. 

The  values  found  from  these  relations  would  serve  as  checks 
upon  the  computation. 

The   computations   may  also   be   checked   by  Euler's  and 
Legendre's  verification  formulas  : 
sin  (36°  +  A)-  sin  (36°  -A)-  sin  (72°  +  A)+  sin  (72°  -  A) 

=  sin  A. 
cos  (36°  +  ^)  +  cos  (36°  -A}-  cos  (72°  +  ^)  -  cos  (72°  -  A) 

=  cos^. 

81.   Approximate  Assumptions.  —  It  can  be  shown  that 
tanV'  -  arc  1"  =  0.00000  00000  00000  04  ; 
arc  1''  -  sin  1"  =  0.00000  00000  00000  02  ; 
tan  1"  -  sin  1"  =  0.00000  00000  00000  06. 
Hence  we  may  assume  that 

siiil''  =  taiil"  =  arcl".  (1) 

In  the  whole  circumference  of  a  circle  there  are  1296000", 
so  that  the  error  due  to  placing  arc  V  =  sin  1"  in  finding  the 
circumference  of  a  circle  with  a  radius  of  unity  will  be  only  2^ 
units  in  the  eleventh  decimal  place. 

In   the    computation    of    elliptic    orbits    there    occurs   the 

equation  M=  U  —  esinU, 

where  M  and  U  are  expressed  in  circular  measure.  If  M"  is 
the  number  of  seconds  in  the  angle,  M=  M"  SircV\  and  ap- 
proximately   M=  M"  sin  1^'  and  U  =  U"  sin  1". 

Hence  the  equation  may  be  written 

M"  =  E" ^sinJS'. 

sm  1" 

*  Ex.  3,  Art.  69,  and  Ex.  8,  Art.  71. 


FUNCTIONS  OF   SEVERAL  ANGLES.  86 

Another  assumption  that  is  often  made  is  that  for  small 
angles  Sinn"  =  n  sin  1".  (2) 

The  error  introduced  is 

for  1',  w"  =  60",       error  =  +  0.00000  00000  04 ; 

for  1°,  n"  =  3600'S  error  =  +  0.00000  09. 
Thus,  if  sin  a  =  0.4  sin  2°,  we  should  have,  since  a  must  be 

small,  a"  sin  1"  =  0.4  sin  2°  or  a"  =  ^liiHl^!. 

sm  1'^ 

82.    Transform  the  First  Member  into  the  Second  (or  last) 
in  the  following  examples  ; 

.      cos  a  —  sec  a         a  <>  ^        /^        t  ^  IN 

1.  =  4  cos^  1  a  (cos2  i  a  —  1). 

sec  a  2     V         2  J  . 

The  first  member  contains  the  angle  a  and  the  second  J  a  ; 
hence  we  must  change  the  angle. 

1_ 

=  cos2  a  -  1  =  (2  cos2 1  a  _  1)2  _  1 


cos  a  — 

cos  a 

cos  a         ^4  gQg4 1 «  -  4  cos2  i  «  =  4  cos2  J  a  (cos^  |^  «  -  1). 

2.   cosec2a  +  cot2a  =  cota.  8.   cosec 2  a  -  cot 2  a  ^  ^^^, ^ 

cosec  2  a  4-  cot  2  a 

4.  cot  a  —  tan  a  =  2  cot  2  a. 

We  may  either  reduce  the  expression  as  far  as  possible  before 
changing  the  angle,  or  change  the  angle  and  then  reduce. 

^  ^    cos  a      sin  a      cos^  a  —  sin^  a        cos  2  a        o  ^  4-  o 

(a) =  — : =  - — : — -—  =  2  cot  2  a. 

sin  a      cos  a         sin  «  cos  a         ^  sin  2  « 

,j.    1  +  cos  2  «     1  -  cos  2  «  _  2  cos  2  «  _  g  ^^^  g  ^ 
sin  2  a  sin  2  a  sin  2  a 

Note. — Avoid  radicals  if  possible. 

6.   sec  a  cosec  o  =  2  cosec  2  a.  8.   cot  J  ^  4-  tan  J  ^  =  2  cosec  6. 

6.    (sin  \d  -\-  cos  J  0)2  =  1  -i-  sin  d.  9.   sin  x  —  2  sin^  x  =  sin  x  cos  2  x. 

l-tan^^^^  1^,    ^(secg  +  sec2g)=    l+tan^^  . 

l  +  tan2jv  ^^  ^     (l-tan2je)2 


S6             PLANE   AND  ANALYTICAL  TRIGONOMETRY. 

.  -        2  tan  ^  V    _   .  13.    1  +  tan  x  tan  J  x  =  sec  x. 

^  14.   ^  (1  +  tan  i  o)2  =  i±^HLi^. 

opf»2  fl  1  +  COS  a 

12.  _§^1_^  =  sec  2  ^. 

2  —  sec2  ^  15.    tan  J  a  4-  2  sin^  J  o  cot  a  =  sin  a. 

16.   «i"^a-tan^^)/ L_^_^ 1 \^sin2x. 

sec^x  Vcosx  — sinx     cosxf  sinx/ 


17.    (1  -  tan2  d)  sin  0  cos  ^  =  cos  2  e-J- 


—  cos  2  g 
+  cos  2  ^ 


18.    l  +  ^^I^=sec2a.  20.   ^-^^l±22l2J:l  =  cot^  6. 

1  —  tan^  a  sec  ^  4-  cos  ^  —  2 

-g    cosg— sing_l— sin2g_    cos2g  21  *  tan  g  ^  "^  ^^  ^  ^  —     ^^"  ^ 

cos^+sin^""    cos2g  "~l4-sin2g*  *             1  -  tan  J  g~  1  -  sin^ 

22.    sec  2  a  +  tan  2  a  +  1  =         ^ 


1  —  tan  o 


23.    (vTTsIna  -  Vl -sina)2  =  4sin2  Jo. 


t- 


24.  (Vl  +  sin  a  +  VT— sino)2  =:4cos2  Ja. 

25.  2  sin  ^  -  sin  (^  -  5)  -  4  sin  A  sin2  J  5  =  sin  (^  4-  -B). 

26.t  cos  (36°  +  ^)  +  cos  (36°  -  ^)  -  cos  (72°  +  ^)  -  cos  (72°  -A)=  cos  ^. 
27.t  sin  (36°  +  A)-  sin  (36°  -A)-  sin  (72°  +  ^4)  +  sin  (72°  -A)z=  sin  ^. 

28.  ?ilL^±iHLl^  =  cotJx. 
cos  X  —  cos  2  X 

29.  1  +  cot2  ^v  = 


sin  V  tan  J  v 

30.   tannv(l +  cot2Av)8=-§ — 
sin^t) 

Q  J     sin  a  cos  ^  g  —  2  cos  a  sin  ^  a  _  „  ^^ga  i 
2  sin  J  a  —  sin  a  ^ 

32    tan2  ^  x  +  cot2  j^  x  __  _  1  +  cos2  x 
tan2  1  X  —  cot2  J  X  2  cos  x 


33.   Given  tan  J  v  =a/    "^  ^  tan  J  ^,  show  that 
'1  —  e 


1  —  e  cos  ^ 


(1  +  e)  cos2  J  V  +  (1  _  e)  sin2  J  v         1  -  e* 
34.  tan  (45°  +  ^)  -  tan  (45°  -A)  =2  tan  2^. 
tan  45°  +  tan  A        tan  45°  —  tan  A 


(a) 


1  —  tan  45°  tan  ^      1  +  tan  45°  tan  A 

_  1  +  tan  A  _  1  —  tan  A  _    4  tan  J.    _  2  tan  2  ^ 
1  —  tan  A      1  +  tan  A      1  —  tan^  A 


*  After  substituting,  multiply  both  numerator  and  denominator  by  the 
quantity  sin  0—  1  +  cos  d. 
t  cos  36°  =  Kl  +  v^). 


FUNCTIONS  OF  SEVERAL   ANGLES.  87 

.,.    1  -  cos(90°  -f  2  A)  1  -  co8(9Q°  -2  A) 

^  ^       sin  (90°  +  2  A}  sin  (90°  -  2  ^) 

1  +  sin  2  ^  1  -  sin  2  ^      2  sin  2  A     o ..      o  a 

COS  2  A  cos  2  ^          cos  2  A 

35  tan  (45°  +  M)  +  tan  (45°  -  ^  A)  _ 

^^  tan  (45°  +  M)  -  tan  (45°  -  M)  ~ 

36.  tan  (45°  +  ^)  -  cot  (46°  +  ^)  =  2  tan  2  ^. 

37.  tan2  (45°  +  6)+  cot2  (45°  +  ^)  =  2  +  4  tan^  2  $, 

38.  tan2  (450  +  „)  -  cot2  (45°  +  o)  =  4  tan  2  o  sec  2  a. 

jg    tan  (45°  +  ^6)  _l  +  8md 
tan  (45°  -  ^  ^)      1  -  sin  tf' 

^^40.  tan^tan(45°  +  J^^)=     ®^^^ 


1  —  sm  ^ 

41.  cot  (45°  -  J  a)  -  tan  (45°  -  J  a)  =  2  tan  o. 

42.  tang(45"4-^a)  =  ^  +  ^"^". 

1  —  sin  a 

43.  tan  (45°  +  6)+  tan  (45°  -  ^)  =  2  sec  2  d, 

44.  l-tan2(45°-^)^^.^^,^ 


iiK    ^      />ico  .  1    .l  +  tanArc     l+sino; 
45.   tan  (45°  +  J  x) ^^  — 


1  +  tan2  (45°  -  6) 

IJ 

1  —  tan  ^x     1  —  sin  a; 

tan  (45°  +  ^x)      _  1  +  sin  a; 

***•   1  +  cot2  (45°  +  ^x)  -  *°°^^  1  -  sinx* 

47.   sin  (45°  -  J  ^)  +  cos  (45°  -  J  e)  =  V2  cos  J  ^  =    ^^^^      . 

VI  -  cos  e 


CHAPTER  VI. 

TRIGONOMETRIC    EQUATIONS. 

83.  One  Equation  Containing  Multiple  Angles.*  —  Change 
the  equation  so  that  it  shall  contain  a  single  angle,  and  then 
proceed  as  in  Art.  52. 

1.  cos  3  a;  =  sin  1x  ;  find  x.     (See  Ex.  8,  Art.  71.) 

4  cos^  x—K>  cos  a;  =  2  sin  x  cos  x . 
.  * .  cos  2:  (1  —  4  sin^  2:  —  2  sin  rr)  -  =  0. 
.  • .  cos  a;  =  0,  giving  x  =  90°  and  270°  ; 
and     1  —  4  sin^  x  —  2  sin  x  =  0,  giving  sin  x  =  ^  ( V5  —  1) 
and     sin  a;  =  -  ^  ( V5  +  1),  or  a:  =  18°,  162°,  234°,  306°. 

2.  cos2 e  +  cos^  =  -  1  ;  find  6.  Ans.  90°,  270°,  120°,  240°. 

3.  cot2  0  +  tan 0  =  -  f  V3  ;  find  d.  Ans.  150°,  330°,  120°,  300°. 

4.  C0S2.X  +  sinx  ^  +  1 ;  find  x.  Ans.  0°,  30°,  150°,  180°. 

5.  sin  3  X  +  sin  2  X  =  sin  x  ;  find  x.  Ans.  0°,  180°,  60°,  300°. 

6.  tan  2  X  =  -  2  sin  X  ;  find  x.  Ans.  0°,  60°,  180°,  300°. 

7.  tan2xtanx  =  +  1  ;  find  x.  Ans.  30°,  150°,  210°,  330°. 

8.  tan2  X  tan  2  X  +  2  tan  X  =  +  V3  ;  find  x.  Ans.  30°,  120°,  210°,  300°. 

9.  sin  4  2r  -  2  sin  2  0  =  0  ;  find  z.  Ans.  0°,  90°,  180°,  270°. 

The  equation  may  sometimes  be  solved  by  the  use  of  the 
equations  of  Art.  73. 

10.  cos  3  a;  —  sin  2  a;  =  0 ;  find  x. 

cos  3  a;  —  sin  2  a;  =  sin  (90°  +  3  a;)  —  sin  2  a; 

=  2  cos  (45°  +  f  a;)  sin  (45°  +  1  a:)  =  0. 
cos (45°+  fa;)  =  0  gives  45°+  fa:  =  90°,  270°,  450°,  630°,  810°, 
or  x=  18°,  90°,  162°,  234°,  306°. 

sin (45°  +  ia:)=  0  gives  45°  +  la;  =  0°,  180°, 
or  a;  =  -90°  and  270'' . 

*  See  Art.  52  for  the  solution  of  equations  wlien  only  one  angle  is  involved. 

88 


TRIGONOMETRIC   EQUATIONS.  89 

11.  cos  9  —  cos  3  e  =  sin  2  fl ;  find  6  by  both  methods. 

Ans.  0°,  30^  90°,  150°,  180°,  270°. 

12.  sin  3  ^  +  sin  2  ^  -f  sin  ^  =  0 ;  find  0  by  both  methods. 

Ans.  0°,  90°,  120°,  180°,  240°,  270°. 

13.  cos2  0  =  sin^;  find  d  by  both  methods.  Ans.  30°,  160°,  270°. 

14.  cos6^-cos3  0  +  sin^  =  O;  findfl.  ^ns.  0°,  180°,  (2  n  +  J  ±  ^) -. 

4 

16.    sin  5  0  +  sin  3  ^  +  2  cos  0  =  0  ;  find  6.  Ans.  90°,  270°,  (2  n  +  f )  -. 

4 

16.  sin  (60°  -X)-  sin  (60°  +  x)  =  +  J  \/3  ;  find  x.  Ans.  240°,  300°. 

17.  sin  (30°  +  X)  -  cos  (60°  +  a;)  =  -  ^  V3  ;  find  x.  Ans.  210°,  330°. 

18.  cos  4;?  -  cos  2  ;?  =  0  ;  find  2.  Ans.  0°,  60°,  120°,  180°,  240°,  300°. 


84.   Find  r  and  <t>  from  the  Equations 


a  and  b  being  known. 

r  sin «(» =  a, 

r  cos  <!>  =  &, 

(1) 

(2) 

(1)^(2)  gives 

tan<^=:^. 

0 

(3) 

From  (1)  and  (2) 

r-     ^     - 

h 

(4) 

s'm(f) 

1.  Find  r  and  <t>  when  loga=0.47141,  and  log  6=0.63927  n,  r  being  positive. 

log  (r  sin  0)=  log  a  =  0.47 141  (1) 

logsin0  =  9.74972  (5) 

logcos0  =  9.91758n  (6) 

log  (r  cos  <p)  =  log  6  =  0.63927  n  (2) 

(l)-(2)  =  log  tan  <^  =  9.8.3214  w  (3) 

0  =  145°48'.4  (4) 

(l)-(5)  =  (2)-(6)=logr  =  0.72169  (7) 

r  =  5.2685  (8) 

The  numbers  on  the  right  indicate  the  order  in  which  the  quantities  are 
found.  If  the  two  values  of  log  r  had  differed,  we  should  have  taken  that  found 
from  log  cos  <p,  as  a  small  error  in  log  tan  <p  would,  for  this  value  of  </>,  affect  the 
logarithmic  cosine  less  than  the  logarithmic  sine.  The  angle  </>  is  placed  in  the 
second  quadrant,  since  r  cos  (f>  is  negative  and  r  sin  <f>  positive,  r  being  considered 
positive. 

2.  Find  r  and  <p  when  log  a=0. 46843  n,  and  log  6=0.43742,  r  being  positive. 

Ans.  <p  =  312°  57'.4  ;  r  =  4.0178. 

3.  Find  r  and  <p  when  log  a  =  1.46444  w,  and  log  6  =  1.86903  n,  r  being 

positive. 

Ans.  <t>  =  201°  30'.0 ;  r  =  79.497. 


90  PLANE  AND   ANALYTICAL   TRIGONOMETRY. 

85.    Find  r,  <|>,  and  6  from  the  Equations 


r  cos  <t>  cos  9  =  a,  ] 

(1) 

r 

sin  4>  cos  0  =  6, 

(2) 

r 

a,  h,  and  o  being  known 

sine           =c, 

(3) 

(2)-^(l)gives 

tan<f)  =  -- 

(4) 

From  (1)  and  (2), 

rcos^-     ^     =     ^    . 
cos  (j)      sin  ^ 

(5) 

From  (3), 

r  sin  ^  =  c. 

(6) 

(6)^(5)  gives 

tan^-^'^''^*-^"^^ 
a               6 

i. 

(7) 

From  r5)  and  (6), 

a                      b 

c 

/^«^ 

cos  <^  cos  ^      sin  </>  cos  ^      sin  0 

1.  Given  loga  =  0.46472,  log6  =  0.72413  n,  logc  =  0.62817,  find  r,  0,  and 
df  d  being  numerically  less  than  90"^,  and  r  being  positive. 

log  (r  cos  «^  cos  e)  =  log  a  =   0.46472  (1) 

log  cos  (f)  =  (9.68314)    Only  as  a  check.  (5) 

log  sin  0=    9.94256  w  (5) 

log  (r  sin  0  cos  d)  =\ogb=    0.72413  n  (2) 

(2)  -  (1)  =  log  tan  0  =    0.25941  n  (3) 

0  =  298°  49'.4  (4) 

(2)-(5)  =  (l)-(5)=log(rcos^)=    0.78157  (6) 

log  cos  ^=    9.91291  (10) 

log  sin  d  =  (9.75951)  Only  as  a  check.  (10) 

logc  =  log(rsin^)=    0.62817  (7) 

(7)  -  (6)  =  log  tan  ^  =  9.84660  (8) 

0=  35°  5'.  1  (9) 

(6)  -  (10)  =  (7) -(10)=  log  r=  0.86866  (11) 

r=  7.3903  (12) 

The  angle  0  is  placed  in  the  fourth  quadrant,  since  r  cos  6  is  positive,  and 
therefore  cos  0  must  be  positive  and  sin  0  negative,  r  cos  0  cos  0  being  positive 
and  r  sin  0  cos  0  negative. 

2.  Given   log  a  =  0.26903  n,  log  &  =  0.32426,   log  c  =  0.36903  w,   find  r,  0, 
and  0,  r  being  positive  and  0  numerically  less  than  90°. 

Ans.  0  =  131°  22'.0 ;  0  =  -  39°  45'. 6  ;  r  =  3.6572. 

3.  Given   log  a  =  9.43942  w,    log  b  =  9.40403  w,    log  c  =  9.56700  w,    find   r, 
0,  and  ^,  r  being  positive  and  0  numerically  less  than  90°. 

Ans.  0  =  222°  40'.1 ;  d  =  -  44«  36'.4 ;  r  =  0.525425  or  0.52544. 


TRIGONOMETRIC   JJQUATIONS.  91 

86.  Find  ^  from  the  Equation 

a  sin<|)  +  6  cos  4»  =  c  (1) 

by  formulas  adapted  to  logarithmic  computation,  a,  J,  and  c 
being  known. 

Let  M  be  an  auxiliary  angle  and  m  a  positive  constant, 
so  that 

m  cos  M=b.  J  ^  -^ 

The  angle  M  is  always  possible,  for  we  have,  by  division, 

tanif=i  (3) 

and  since  the  tangent  may  have  any  value  between   +  oo  and 
—  00,  there  will  always  be  some  angle  whose  tangent  is  equal 

to    Y*     Also,  squaring  and  adding  Eqs.  (2),  we  have 

0 

m^  sin^  il[f  +  m^  cos^  M=  m^=  a^  -\-  h\ 

or  m  =  Va^  +  b'^.  (4) 

Therefore  the  assumptions  in  (2)  are  always  possible,  since  M 
and  m  will  be  real  quantities  if  a  and  h  are  real. 
Substituting  (2)  in  (1),  we  have 

m  sin  M  sin  <f)  -\-  m  cos  M  cos  <f>  =  c^ 

\     or  m  cos  ((/>  —  Jf )  =  c.  (5) 

Hence,  from  (2)  find  M  and  m  by  the  method  of  Art.  84 ; 
from  (5)  find  <\)  —  iJf  (two  values  <  360°),  and  thence  find  </>. 

1.  Find  0  when  2  sin  ^  —  3  cos  0  =  1. 
Ans.  M=  146°  18'.6  ;-  0  =  220°  12'. 5,  or  72°  24'. 7. 

2.  Find  <t>  when  2  sin  0  +  4  cos  0  =  —  3. 
Ans.  ilf=26°33'.9;  0  =  158°41'.8,  or  254°26'.0. 

87.  Find  <()  from  the  Equation 

atan<|>  +  &cot<|>  =  c 

by  formulas  adapted  to  logarithmic  computation,  a,  5,  and  c 
being  known. 


92  PLANE   AND   ANALYTICAL   TRIGONOMETRY. 

Substituting  for  tan  <t>  and  cot  </>  in  terms  of  sin  (f>  and  cos  <^, 
we  have,  after  reducing, 

(a  —  6)  cos  2(l>  -\-  c  sin  2(f)  =  a  -{-  b. 

Let    msinM=a  —  b^ 
m  cosilff  =  c. 


\     .-.  wsin  (il[f  4- 2(/>)  =  a4- ^. 


1.  Find  0  when  2  tan  0  —  cot  0  =  —  3. 

^ns.  ilf=135°;  0  =  15°41'.O,  119M9'.0,  195°41'.0,  299°  19'.0. 

2.  Find  0  when  tan  0  +  3  cot  0  =  —  2  VS. 

^ns.  M  =  210°  ;  0  =  120°  or  300°. 

88.    Find  <}>  from  the  Following  Equations,  a  and  a  being 
known: 

(a)    sin  (<^  -f-  «)  =  «  sin  </>.  (1) 

Expanding,  sin  <^  cos  a  +  cos  <^  sin  a  =  a  sin  <^. 
.  • .   sin  (^  (a  —  cos  a)  =  cos  <^  sin  a, 

,        ,           sin  a  ,o\ 

.-.  tan<f>  = (2) 

a  —  cos  a 

Eq.  (2)  is  not  adapted  to  logarithmic  computation.     But 

from  (1)  we  have 

sin  (0  -f  «)  _  a 
sin<^  i 

and,  by  composition  and  division, 

sin  (<^  +  a)  +  sin  <^  _  a  +  1 
sin  ((/)  +  «)—  sin  <^      a  —  V 

and  this,  from  the  equations  of  Art.  73,  becomes 

tan  (<^  -f  1^  a)  _  a  4- 1 
tan^^a  a  —  l' 

or  tan  (<^  +  1  a)  =  ^  "^     tan  \  a.  (3) 

Let  tan/3  =  a,  and  note  that  tan  45°  =  1. 

^      ^^       .    ^      tan /3  +  tan  45°  ^      , 
...  tan(c/>  +  J«)=^-^^^^-^^^^^tanla 

sin(y8  +  45°)^      , 

=    .    \ri TF^  tan i  a. 

sm  (/S  —  45°)         2 

.-.  tan(</)-fla)=cot(/3-45°)tanla.         (4) 


TRIGONOMETRIC   EQUATIONS.  93 

(6)        COS  (<^  -f  a)  =  <«  cos  <l>. 

.  • .  tan  (<^  -f  J  «)  =  tan  (45°  —  /S)  cot  J  a,  if  tan  /8  =  a. 
(<?)        sin  (a  —  (f)^=  a  sin  <^. 

.  • .  tan  (<^  —  I  «)  =  tan  (45°  —  y8)  tan  J  a,  if  tan  y9  =  a. 
(^)        sin  (</>  +  a)  =3  a  cos  </>. 

.-.  sin(<^  +  a)=«sin(90° +<^); 
.-.  tan(4'5°  +  </)  +  |«)  =  cot(45°-)8)tan(45°-^a),  if  tan)S=a. 
(g)        cos  (</)  +  a)  ==  a  sin  <^. 

.-.  cos(</>  +  «)=acos(90°  -  <^); 
.  • .  tan  (<^  4-  i  a  -  45°)  =  tan  (45°  -  ff)  cot  (45°  +  J  a),  if  tan  0  =  a. 

Note.  — The  equation  a  sin  (0  +  a)  =  a'  sin  (0  +  a')  and  similar  equations 
may  be  solved  by  expansion,  the  solution  of  the  given  equation  being 

,„     ,      a' sin  a'  —  a  sin  a 

tan  0  = -. 

a  cos  a  —  a'  cos  a' 

A  solution  adapted  to  logarithmic  computation  may  be  found  by  the  method 
of  this  article,  giving 

tan  [0  +  ^  (o  +  a')]  =  cot  (j8  -  45°)  tan  K«  -  «')»  i^  tan  &  =  —- 


89.    Find  <t>  from  the  Following  Equations,  a  and  a  being 
known : 

(a)    sin  (<^  +  a)  sin  <l>  =  a. 
From  (8),  Art.  72, 

cos  a  —  cos  (2  <^  +  a)  =  2  a. 
. • .  cos (2  <^  +  a)  =  cos  a  —  2a.  (1) 

Let  tan  /3  =  ^-^.  (2) 

sin  a 

^n  ,  ,     ^  '       .       a     COS  a  COS  j8  — sin  a  sin  y3 

.*.   cos(2<f>  +  a)  =  cosa  — sinatan)^= — ^• 

cos)8 

■  •■  coa^24>  +  a)=^°^("+/\  (3) 

cosp  ^  ^ 

(5)    sin  (a  —  <^)  sin  (\>  =  a. 

.  • .   cos  (a  —  2  <^)  —  cos  a  =  2  a  ; 

.  • .   cos  (a  —  2  <^)  =  cos  a  +  2  a. 

^         o  JL\      cos(a  — /3)    ./.  .       o        2  a 

•  o  cos  («  —  2  0)  = ^^ — — ^-^ ,  if  tan  p  = 

cos  p  sin  a 


94  PLANE   AND   ANALYTICAL  TRIGONOMETRY. 

(c)    sin  (0  +  a)  cos  cf)  =  a. 

.  • .  sin  (2  </>  +  a)  -t-  sin  «  =  2  <i. 

•    ^o  J.   ,     \      o          •           sin(yS  — a)   .0,       o       2  a 
.  • .  sin  (2  </)  +  a)  =  2  a  —  sin  a  = i^i-— — ^,  if  tan  ^  = 

cos  p  cos  a 

(^d)    cos  (</)  H-  a)  cos  <f>  =  a. 

.  • .   cos  (2  (^  +  a)  +  cos  a  =  2  a. 

,  • .  cos  (2  <f)  +  a)  =  2  a  —  cos  a  = ^^ — '-f^ ,  if  tan  p  = 

cos  /3  sin  a 

(e)    cos  (<^  +  «)  sin  (j)  =  a, 

.*.  sin(2<^  +  a)  =  2a  +  sina  = ^^ — tt^^  ^^  tanyQ= 

cos  p  cos  a 

90.    Find  cj)  from  the  Following  Equations,  a,  a,  and  «'  being 
known  : 

(a)    tan  (</>  +  «)=«  tan  (f>. 

tan  (<j)  +  «)  _  g .    .    tan  (<^  +  «)  +  tan  <^  _  a  +  1 . 
tan^  l'         tan  ((/)  +  a)  —  tan  (/>      a  —  l' 

sin  (2  <^  +  ct)  _  g  +  1 
sin  a  a  —  1 

Let  tan/8  =  a;  (2) 

.-.  ^  =  cot(yS-45°), 

and  sin  (2  <^  +  «)  =  cot  (/3  -  45°)  sin  a.  (3) 

Find  /3  from  (2)  and  2  <^  +  «  from  (3). 
(V)  tan  ((^  +  a)  =  a  cot  <f>. 

.  • .  cos  (2  <^  +  «)  =  tan  (45°  —  yS)  cos  a,  if  tan  0  =  a. 
(c)    cot  (a  —  <^^=  a  cot  <^. 

.  • .   sin  (2  (/)  —  a)  =  tan  (/3  —  45°)  sin  a,  if  tan  /3  =  a. 
(c?)  cot  (</)  +  a)  =  a  cot  (<^  —  a). 

.  • .   sin  2  (/>  =  cot  (45°  —  yS)  sin  2  a,  if  tan  fi  =  a. 
(e)  tan  ((^  +  a)  =  a  tan  (</>  +  «'). 
. • .   sin  (2  <^  +  a  +  «')  =  cot  (y8  -  45°)  sin  (a  -  «'),  if  tan  yS  =  a. 

(/)  cot  ((/>  +  a)  =  a  cot  (<^  +  a')- 
.• .   sin  (2  </>  +  a  4-  "0  =  cot  (/S  -  45°)  sin  (a'  -  a),  if  tan  /S  =  a. 

(^)   cot  (<^  +  a)  =  a  tan  (</>  +  «')• 
.  • .   cos  (2  (^  +  a  +  a')  =  tan  (y3  —  45°)  cos  (a  —  a'),  if  tan  yS  =  a. 


(1) 


TRIGONOMETRIC   EQUATIONS.  95 

91.    Find  ^  from  the  Following  Equations,  a,  a,  and  a'  being 
known :  * 

(a)  tan  (<^  -|-  «)  tan  <f)  =  a. 
.  • .   sin  (</>  -}-  a)  sin  </>  =  a  cos  (</>  +  a)  cos  ^. 
From  the  equations  of  Art.  72,  we  have 

—  cos  (2  </>  +  a)  +  cos  a  =  a  cos  (2  (f) -{•  a) -\- a  cos  a; 

.-.  cos  (2</).H- a)  =  --^^^— cosa. 
1  +  ^ 
Let  tan  fi  =  a. 

.  • .  cos  (2  </>  +  «)  =  tan  (45°  —  ;8)  cos  a. 
(V)    tan  (<^  +  a)  cot  </>  =  «. 

.  • .  sin  (2  </)  +  «)=  cot  (y8  —  45°)  sin  a,  if  tan  ^  =  a. 

(c)  tan  (<^  +  a)  tan  ((f>  —  a)=  a. 

.  • .  cos  2  <^  =  tan  (45°  —  y8)  cos  2  a,  if  tan  P=a. 

(d)  tan  (^  +  a)  cot  (<^  +  «')  =  a. 

.  • .   sin  (2  </>  +  «  +  «0  =  cot  (y8  —  45°)  sin  (a  —  a'),  if  tan  ^  =  a. 


92.    Find  r  and  <t>  from  the  Following  Equations,  a,  h,  a,  and 
/9  being  known  : 

r  sin  (<^  +  «)  =  a,  I  (1) 

rcos(<^  +  /3)=  J.  J  (2) 

sin  (d)  -f  a)       <*  i    •     ^  j    ,     \  ^  i    ,    o\ 

.  • .  5  sin  <^  cos  a-\-h  cos  <^  sin  a  =  a  cos  <\>  cos  ^  —  a  sin  <^  sin  yS. 

.  • .   5  sin  (\>  cos  a  4-  a  sin  <^  sin  p  =  a  cos  <^  cos  /S  —  b  cos  <^  sin  a. 

.  • .  sin  (f)  (b  cos  a-{-  a  sin  y8)  =  cos  </>  (a  cos  /3  —  5  sin  a). 

.  • .  tan  6  = — r,  (3) 

b  cos  a  -\-  a  sin  p 

and  r  = = (4) 

sin(<^  +  «)      cos(</>  +  y8) 

The  quadrant  of  (j)  will  be  determined  by  the  sign  assigned 
to  r. 

*  The  method  of  Art.  90  may  be  used,  since  tan  x  = and  cot  x  = • 

cot  X  tan  X 


96 


PLANE   AND   ANALYTICAL   TRIGONOMETRY. 


1.  If  r  sin  (0  +  a)  =  a,  and  r  sin  (0  +  /3)  =  6,  show  that 

6  cos  a  —  a  cos/3 

2.  If  r  cos  (0  +  a)  =  a,  and  r  cos  (0  +  /3)  =  6,  show  that 

♦«^ -J.      rt  cos /3  —  ?>  cos  a 

tan  0  = 

as>'\n3  —  b  sin  a 


;:i 


93.    Find  r  and  <|)  from  the  Following  Equations,  a,  5,  a,  and 

/3  being  known,  and  the  formulas  derived  being   adapted   to 
logarithmic  computation : 

r  sin  (</)  +  «)  =  a, 

r  sin  {(f)  +  /3)=b 

(1)  4-  (2)  =        r  [sin  (<^  +  «)  +  sin  (<^  +  yg)]  =  «  +  5 ; 

.-.   2rsin[<^  +  l(a  +  y3)]  cos  |  (a  -  y8)=  a  +  J. 

(1)  -  (2)  =        r  [sin  ((^  +  «)  -  sin  ((^  -f  ^)]  =  «  -  ^  ; 

.  • .    2  r  cos  [(^  +  ^  (a  +  /S)]  sin  |^  (a  -  y8)  =  a  -  ^>. 

From  (3)  and  (4),  we  have 

a  +  b 


0) 
(2) 


r  sin  [<^  +  K«  +  ^)] 
rcos[<^4-K"  +  ^)]  = 


2cos-|-(te-y3)' 


2 

a  —  h 


2sini(«-y8)    J 

from  which  r  and    </>  4-  |  («  +  y8)  are  found  by  the  method  of 
Art.  84. 

1.    If  r  cos  (0  4-  a)  =  a,  and  r  cos  (0  +  /3)  =  &,  show  that 

a  -\-h 


rcos[0+i(«  +  ^)]  = 
rsin[0  4-Ka  +  '8)]  = 


2  cos  I  (a  —  ii)* 

h  —  a 


2  sin  \{a-  fi) 

2.    If    r  sin  (0  4-  a)  =  a,    and    r  cos  (0  +  y8)  =  &,    show  that    by    placing 
cos  (0  +  /3)  =  sin  (90"^  +  0  +  3)  we  may  obtain 

a  +  & 


rsin[0  +  45°  +  Ha  +  )8)]  = 
rcos[0  +  45°  +  Ka  +  i8)] 


2  cos  [45°-  Ka-^)] 
h  —  a 


2  sin  [45°-  K«-^)] 
3.   Find    r   and    0   when    rsin  (0  +  100°)  =  2,    and  r  sin  (0  +  200°)  =  3, 

r  being  positive. 

Ans.  0  =  290°  28'.  4  ;  r  =  3.9436. 


CHAPTER   VII. 


OBLIQUE   PLANE  TRIANGLES. 


94.  It  has  been  shown  in  Geometry  that  a  triangle  can  be 
constructed  when  three  elements,  one  being  a  side,  are  known. 
If  the  three  angles  only  are  given,  there  will  be  an  infinite 
number  of  triangles  satisfying  the  conditions  of  the  problem, 
since  the  data  determine  the  shape  and  not  the  size  of  the 
triangle. 

We  also  know  that  in  any  triangle 

(1)  The  sum  of  the  three  angles  is  180°. 

(2)  If  one  angle  is  90°,  the  sum  of  the  other  two  is  90°. 

(3)  The  greater  side  is  opposite  the  greater  angle,  and 
conversely. 

(4)  Any  side  is  less  than  the  sum  of  the  other  two. 

95.  The  Sine  Proportion.  —  The  sides  of  a  triangle  are  to  each 
other  as  the  sines  of  the  opposite  angles. 


In  Fig.  52,  p  =  a  sin  7  ;  p  =  c  sin  a. 

.*.   a  sin  7  =  <?sin  a. 


a      sm  a 


or 


c      sin  7 

CROCK.  TRIG.  — 7  97 


sin  a      sin  7 


(1) 

(2) 


98 


PLANE  AND  ANALYTICAL   TRIGONOMETRY. 


In  Fig.  53, 

J)  =  a  sin  y'  =  a  sin  (180°  —  7)  =  a  sin  7,  and  p  =  c  sin  a. 

.'.  a  sin  7  =  c  sin  «,  as  before. 

In  the  same  way,  by  drawing  a  line  perpendicular  to  AB 

from  C  (Figs.  52  and  53),  we  can  show 

that 

a  h 

d =  - — -' 

sin  a      sm  p 

5  '   *  sin  a     sin  p     sin  -y'  ^  ^ 

true  for  both  acute  and  obtuse  angled 
triangles. 

Note.  —  The    constant    quotient    — ^    is 

sin  a 
called  the  modulus  of  the  triangle,  and  is  equal 
to  the  diameter  of  the  circumscribed  circle. 


Fig.  54. 


For,  in  Fig.  54,  c  =  AB  =  2  B  sin  ADD  =  2  B  sin  I  AOB  =  2  B  sin  7. 

c 


=  2B. 


sm7 


96.   The  Square  of  Any  Side  of  a  Triangle  is  equal  to  the  sum 
of  the  squares  of  the  other  tivo  sides,  diminished  hy  twice  the 
N    product  of  the  two  sides  multiplied  hy  the  cosine  of  their  included 
angle- 


From  geometry  we  have,  m  Fig.  b^^ 

c^  =  a'^  +  h^-2b  '  DC=  a^  +  b^-2 abcosy.  (1) 

Also,  in  Fig.  b6, 

c^  =  a'^-{-  h^  -\-  2b  -  CD  =  a"^  +  P  +  2ah  cos 7', 
or  c^  =  a^  +  b'^-2abcosy.  (2) 


OBLIQUE   PLANE  TRIANGLES. 


99 


This  relation  may  also  be  proved  as  follows  : 
In  Fig.  55,  b  =  AC=  AD  -\-  DC  =  ecosa  ^  a  cos  7. 

In  Fig.  56,  b  =  AC  =  AD  -  CD  =  ccosa-  a  cos 7' 

=  c  cos  a  -f-  a  cos  7. 
.-.  6  =  (?cos  «  +  a'cos  7. 
.*.  <?cos  a  =  5  —  a  cos  7. 
.  •.  c^  cos^  a  =  a^  cos^  y  +  P—  2ab  cos  7. 

from  (-1),  Art.  95. 


But 

By  addition, 

since  sin^  x  +  cos^  x 


c^  sin^  a  =  a^  sin^  7, 

c2  =  ^2  _|_  52  _  2  ab  cos  7, 
1. 


97.    Case  I.     Given  One  Side  and  Two  Angles  (a,  a,  P). 

Formulas  :  7  =  180° -(a +  13}; 


b  = 


a 


c  = 


sma 
a 


sinyS; 
sin  7. 


Fi&.  57 


sin« 

1.   Solve  the  triangle  when  a  =  3.4356, 
a  =  17°43'.4,  7  =  60^35'.7.      ^ 

.-.  $  =  180°  -  (a  +  7)  =  101°  40'.9. 
(a)   By  natural  functions. 

6  =  a  X  sin  3/-^  sin  o  =  3.4356  x  .97929  ^  .30442  =  11.052. 
c  =  a  X  sin  7^-^  sin  a  =  3.4366  x  .87117  ^  .30442  =    9.8318. 
(6)    By  the  use  of  logarithms. 

log  &  =  log  a  —  log  sin  o  +  log  sin  )3  =  log  a  +  col  sin  a  +  log  sin  3. 
log  c  =  log  a  —  log  sin  a  +  log  sin  7  =  log  a  +  col  sin  a  +  log  sin  7. 

log  a  =  0.53600  log  a  =  0.53600 

col  sin  a  =  0. 51652  col  sin  a  =  0.51652 

log  sin  j8  =  9.99091  log  sin  7  =  9.94010 

log  &  =  1 .  04343  log  c  =  0. 99262 

6  =  11.052  c  =  9.8315 


2.  Solve  the  triangle  when  c  =  54.376,  a  =  103°  3'.2,  ^  =  40°  10'.3. 

Ans.  7  =  36°  46'. 5  ;  6  =  58.591 ;  a  =  88.478. 

3.  Solve  the  triangle  when  a  =  0.14323,  «  =  53°  17 '.3,  p  =  62°  23'. 5. 

Ans.  7  =  64°19'.2;  6  =  0.15832;  c  =  0.16101. 


100 


PLANE   AND  ANALYTICAL  TRIGONOMETRY. 


98.    Case  II.     Given  Two  Sides  and  the  Angle  Opposite  One 
of  them  (a,  c,  a).  — From  the  sine  proportion,  we  liave 


sm  7  =  -  sin  a. 
a 


(1) 


Since  7  is  found  from  (1)  by  means  of  its  sine,  it  may  have 
two  values,  one  in  the  first  and  one  in  the  second  quadrant,  their 
sum  being  180°.  Therefore  there  maj/  be  two  triangles  with 
the  given  elements. 

If  a  is  obtuse,  7  must  be  acute,  since  there  can  be  only  one 
obtuse  angle  in  a  plane  triangle,  and  there  will  be  only  one 
solution.  7 

•  If 'a  is  acute,  and  a  is  greater  than  c,  7  will  be  acute,  since  ' 
a  must  be  greater  than  7,  and  there  will  be  only  one  solution. 


C'^^^.  D  ^.-'  c 


Fig.  58. 


If  a  is  acute,  and  a  is  equal  to  c,  there  will  be  only  one 
solution,  since  the  points  C  and  A  Avill  coincide. 

If  a  is  acute,  and  a  is  less  than  c,  7  will  be  greater  than 
a,  and  therefore  7  may  be  either  in  the  first  or  in  the  second 
quadrant. 

In  order  that  there  may  he  two  solutions^  the  given  angle 
must  he  acute^  and  the  side  opposite  it  must  he  less  than  the 
side  adjacent. 

li  a  =  DB,  the  two  triangles  will  be  coincident,  7  being  90°. 
If  a  is  less  than  DB,  the  triangle  will  be  impossible ;  this  will 
be  shown  in  the  computation  where  sin  7,  found  from  (1^,  will 
be  greater  than  unity. 

If  we  use  primed  letters  to  represent  the  unknown  elements 
of  one  of  the  triangles,  and  unprimed  letters  for  those  of  the 
other,  we  have 


OBLIQUE  PLANE   T'RI ANGLES.. 


aoi 


Formulas  :    sin  7  =  -  sin  «  =  sin  7' ; 
a 


|Q  =  180°-(a  +  7);   ^8'^  180°-(c6  +  7') 

d  _•_    r,        n  CI 

sin  a 


or 


h 


sinyS;  b' 


sm« 


sin/3^ 


sinyS;   h' = ^sin/S'. 


sin  7  sin  7' 

1.    Solve  the  triangle  when  a  =  9.4672,  c  =  14.433,  a  =  1 1°  14'.3. 

In  this  example  a<90'^,  a<c;    . •.  two  solutions, 
log  sin  7  =  log  c+col  a  +  log  sin  a  =  log  sin  y'. 

&  =  180^^  -(a  +  7)  ;  3'  =  180°  -(a  +  7')- 

log  6  =  log  a  +  col  sin  a  +  log  sin  3  =  log  c  +  col  sin  7  +  log  sin  j8. 

log  b'  =  log  a  +  col  sin  a  +  log  sin  ^'  =  log  c  +  col  sin  7'  +  log  sin  n'. 


.-,  lOfrl 


log  c=  1.15936 

col  a  =  9.02378 

lofesin  a  =  9.28979 


log  sin7  =  9.47293 

7=    17°17M 

7'  =  162°  42'.9 

.-.  )8  =  151°28'.6 

i8'=      6°   2'.8 


log  a  =  0.97622 
col  sin  a  =  0.71021 
logsinj8  =  9.67899 

log  6 
b 


1.36542 
23.196 


logc  =  1.15936 
col  sin  7  =  0.^2707 
logsin/3  =  9.67b99 

log  6 
b 


1.36542 
23.196 


log  a  =  0.97622 
col  sin  o  =  0.71021 
log  sinks' =  9.02259 

log  6' =  0.70902 
b'  =  5.1170 


log  c  =  1.15936 
col  sin  7' =  0.52707 
log  sin  3' =  9.02250 

log  6' =  0.70902 
b'  =  5.1170 


t 


2.  Solve  the  triangle  when  a  =  2.4741,  c  =  1.0003,  a  =  69°  14'.  8. 

Ans.  7  =  22°  12'.8  ;  )8  =  88°  32'.4  ;  b  =  2.6449. 

3.  Solve  the  triangle  when  a  =  10.473,  b  =  12.987,  a  =  44°  11 '.3. 

j  iS  =  59°  48'.5  ;  7  =  76°  0'.2  ;  c  =  14.579  ; 
I  &'  =  120°  11'.5  ;  7'  =  15°  37'.2  ;  c'  =  4.0456. 

4.  Solve  the  triangle  when  a  =  0.43477,  b  =  0.40031,  a  =  94°  17'.6. 

Ans.  &  =  m°  39'.6  ;  7  =  19°  2'.8  ;  c  =  0.14228. 

99.   Case  III.      Given  the  Three  Sides  (a,  b,  c). 

(a)  From  Art.  96, 

a^  =  l)^  -\-  c^  —  2  be  cos  a. 

b^  +  c^  —  a* 


.*.  cos  a  = 


2  6c 


(1) 


From  this  equation  we  may  find  a  by  means  of  its  natural 
cosine. 


102 


l^L ANE   AND  ANALYTICAL   TllIGONOMETRY. 


(6)  To  adapt  (1)  to  logarithmic  computation,  subtract  each 
member  from  unity. 

.'.  1  — cosa  =  l ' = ' — = ^^ —' 

2  be  2  be  2  be 

.  .   ^sin  J  a-  26e  -  2Vc 

Let  a-\-b  -\-  e  =  2  8  ; 

.\a-\-b  —  c  =  a-{-b  +  c-2e  =  2s  —  2c=2(s  —  c'); 
.    a-b-]-c  =  a-\-b-{-e-2b  =  2s-2b  =  2(s-b'). 

'   .    sin2iu      2(8-5)2(.-.)_(.-5)(8-.X 

.     .      Dili     ^  W    =  — ^ ' 

where  8  is  half  the  sum  of  the 
three  sides,  and  b  and  c  are  the 
sides  adjacent  to  the  angle. 


(2) 


((?)  Again,  adding  each  member  of  (1)  to  unity, 

52  +^2___a2  ^  2ic^V^^c'^-a^  ^  (b  -f  g)^  -  g^ 
2  6(?        ~  2  5(?  2be         ' 


1  +  cos  «  =  1  4- 


••'^''''^2«-  2Ve  2be 

o  1        s  (s  -  a) 


s  (s  -c) 


ab 


(3) 


(c?)  Dividing  sin^  J  «  by  cos^  |^  a,  we  have 
Similarly, 


2  s(s  -  a) 


(s  —  a)  (s  —  c) 


taii^lP 


(4) 


. 


OBLIQUE  PLANE   TRIANGLES. 


103 


Or 

tan2^a  = 


(8  —  a)(«  —  b) (s  —  c)  _  («  —  a)  (s  —  6) («  —  c) 


6t(«  — a)^ 


(8 -a) 


tan  J  a 


4.V 


(8  —  a)(8  —  b)  (s  —  c) 

8 


Let 


Similarly, 


A 


s  -  «)  (g  -  6)  (g  —  c) 


.•.  taii^a  = 
tanip 


tan^Y  = 


s  —  a 
r 


(6) 


(6) 


The  angles  of  the  triangle  may  be  found  from  (2),  (3),  (4), 
or  (5)  and  (6),  the  computation  being  checked  by 

i-«  +  i^  +  i7  =  90°. 

In  finding  all  the  angles,  (5)  and  (6)  should  be  used. 

Note. — The  tabular  difference  for  tana;  is  greater  than  that  for  either 
sin  a;  or  cos  x,  so  that  a  small  error  in  tan  x  will  affect  the  angle  x  less  than 
would  a  corresponding  error  in  sinx  or  cos  a;.  Hence  the  angles  should  be 
determined  by  means  of  their  tangents  whenever  practicable. 

Again,  when  x  is  less  than  45°,  the  tabular  difference  for  sin  x  exceeds  that 
for  cos  X,  and  when  x  is  greater  than  45°,  the  tabular  difference  for  cos  x  is  the 
greater.  Hence  the  angle  should  be  determined  by  means  of  its  sine  rather 
than  its  cosine  when  the  angle  is  less  than  45°,  and  by  its  cosine  rather  than  its 
sine  when  it  is  greater  than  45^. 

Note.  —  r  is  the  radius  of  the  inscribed  circle.     For,  considering  the 

/\ABC  =  AOAC+AOCB-^AOBA 

=  4£0E,   +^OF+—OD 

2  '  2  2 

=  \  {AC  +BC+  AB)  r  =  sr. 
But,  from  Art.  109, 


A  ABC: 

=  Vs(s 

-a)is 

-b)is- 

c). 

.-.  sr 

=  Vs(s 

-a)(s 

-b){s- 

c). 

-J(- 

a)(s- 

b)(s-c) 

Solve  the  triangle  when  a  =  0.0093146,  b  =  0.0176530,  c  =  0.0095768. 
log  r  =  ^  [log  (s-  a)  +  log  (s  -  6)  +  log  (s  -  c)  +  col  s], 
log  tan  1  a  =  log  r  —  log  {s  —  a),  etc. 


104 


PLANE   AND  ANALYTICAL   TRIGONOMETRY. 


a  =  0.009314G 

6  =  0.0176530 

c  ^  0.0095768 

2  s  =  0.0365444 


log  (s  -  a) 

log  (s  -  h) 

log  (s  -  c) 

col  s 


7.95219  -  10 
6.79183  -  10 
7.93929 
1.73821 


10 


s  =  0.0182722 

log  r2  =  4.42152  -  10 

s-a  =  0.0089576 

log  r=  7.21076 -10 

s-b  =  0.0006192 

.-.  logtan|a  =  9.25857 

s-c  =  0.0086954 

^  a  =  10°  16'.8 

sum  =  0.0365444 

logtan  1)3  =  0.41893 

2  s  =  0.0365444 

^  )8  =  69°  8'.2 

a  check. 

log  tan  ,1  7  =  9.27147 

i  7  =  10°  35'.0 

In  finding  log  tan  \  a,  write  log  r  on  the  margin  of  a  slip  of  paper,  place  it 
above  log(s  — a),  and  write  the  difference  of  the  two  logarithms  opposite 
log  tan  \  a  ;  then  find  log  tan  \  $  and  log  tan  |^  7  in  the  same  way.  Find  s  —  a, 
s  —  6,  and  s  —  c  in  a  similar  manner. 

2.  Solve  the  triangle  when  a  =  32.456,  h  =  41.724,  c  =  53.987. 

Ans.  i  o  =  18°  27'.4  ;  ^  )8  =  25°  16'.3  ;  1  7  =  46°  16'.4. 

3.  Solve  the  triangle  when  a  =  0.14679,  b  =  0.10433,  c  =  0.04796. 

A71S.  ^  a  =  73°  20'.4  ;  ^  i8  =  11°  29'.4  ;  ^7  =  0''  10'.2. 


^ 


100.  Case  IV.  Given  Two  Sides  and  the  Included  Angle 
(bf  Cf  a) .  First  Method.  —  The  sum  of  any  two  sides  of  a  tri- 
angle is  to  their  difference  as  the  tangent  of  half  the  sum  of  the 
opposite  angles  is  to  the  tangent  of  half  their  difference.'    For 

we  have 

b  _  sin  y8 
c      sin  7 
By  composition  and  division, 

b  -\-  c  _  sin  ff  +  sin  7 
b  —  c     sin  y8  —  sin  7 

_2sini(/3  +  7)  cos^(/3-7). 
2  cos  I  (J3  +  7)  sin  ^{^  -y)' 
\Art.  73.) 

tan  I  (P  +  v) 


b  +  c 
b-c 


But 

/3  4-7=180^ 


taii|(p 


Y) 


(1) 


«;  -K/S  +  7)  =  90°-l«.   .-.  tanl(/3  +  7)  =  coti«. 
b 


tani(;Q-7) 


b  +  c 


cot  \  a. 


(2) 


OBLIQUE   PLANE  TRIANGLES. 


105 


From  (2)  we  find  ^(j3-y);  adding  iC^-y)  to  i(l3  +  y), 
we  have  yS,  and  subtracting  J  (/S  —  7)  from  J  (yS  +  7),  we  have 
7.     Then  the  third  side  is  found  from  the  sine  proportion. 

h-c 


Formulas  :    tan  J  (y8  —  7) 

K/^  +  T) 

7 


cot  I  a, 


97  «, 


90° 

K/3  +  7)-K/5 

5  sin  a      ^  sin  a 


7), 
7), 


sin  yS        sin  7 

In  using  (1)  or  (2)  the  greater  side  and  the  greater  angle 
should  be  written  first ;  thus,  if  c  were  greater  than  b,  we 
should  use  c  —  b  and  7-/3  instead  oi  b  —  c  and  /3  —  7.  If  the 
smaller  side  is  written  first,  the  tangent  of  half  the  difference 
of  the  two  angles  will  be  negative,  giving  the  half-difference 
as  an  angle  between  0°  and  —  90°. 

1.    Solve  the  triangle  when  b  =  0.14367,  c  =  0.11412,  a  =  42°  14'.6. 
.-.  ^a  =  21°7'.3;  Ki^  +  7)  =  90°  -  ^a  =  68°52'.7. 
log  tan  ^  ()8  —  7)  =  log  (6  -  c)  +  col  (ft  +  c)  +  log  cot  |  a. 
log  a  =  log  6  +  log  sin  a  +  col  sin  j3  =  log  c  +  log  sin  a  +  col  sin  7. 


b-G  =  0.02955 
6  +  c  =  0.25779 


log  (b  -  c) 
col  (6  +  c) 
log  cot  I  a 

log  tan  K^  -  t) 

K^  +  7) 

/3 
7 


:  8.47056 
0.58874' 
0.41308 

9.47238 
16°31'.7 

68°  52'.  7 

85°  24'.  4 
52°21'.0 


log  5  =  9.15737 
logsin  a  =  9.82755 
col  sin  /3  =  0.00140 

log  a 


8.98632 
a  =  0.096900 


logc 

Ipg^ina 

-^ol  sin  7 


9.05737 

9.82755 
0.10141 


^ 


log  a  =  8.98633 
a  =  0.096902 


2.  Solve  the  triangle  when  a  -  101.47,  c  =  99.367,  /3  =  47^  48'.2. 

Ans.  a  =  67°27'.l  ;  7  =  64°  44'. 7  ;  b  =  81.396  or  81.394. 

3.  Solve  the  triangle  when  b  =  19.937,  c  =  62.475,  a  =  1.30°  9'.4. 

Ans.  /3  =  11°  26'.1  ;  7  =  38°  24'. 5  ;  a  =  76.858  or  76.860. 

101.    Case  IV.     Given  6,  c,  a.     Second  Method.  —  To  prove 
the  equations 

a  sin  1  TyS  —  7)  =  (6  —  (?)  cos  1^  a,  |  (1) 

acosl(/3-7)  =  (6  +  <?)sin^a.   J  (2) 


106 


PLANE   AND   ANALYTICAL   TRIGONOMETRY. 


h 

_sinyS 

. 

h  ^c 

sin  yS  +  sin  7 

c 

Sin  7 

c 

sm  7 

h  + 

c 

c            a 

a 

sin 

/3  + 

sin  7 

sin  7      sin  a 

sin  (^  +  7) 

h-\-  c 

a 

2  sinl(/3  +  7)cos  i(/^  -  7)      2  sini(/S  +  7)  cos  -|-(y8  +  7) 
.-.  acosl-(^-7)  =  (5  +  c)cos}(/3  +  7) 

=  (5  +  (?)  sin  ^  a.  Q.E.D. 

Similarly,  ~    = —. reduces  to 

c  sin  7 

a  sin  J  (yS — 7)  =  (5  —  (?)  sin  l  (/8  +  7)  =  (^  —  0  ^^^  |^  a. 

1.   Solve  the  triangle  when  6=0.14367,  c  =  0.11412,  a=42°14'.0. 
h-c  =  0.02955     (1)       (4)  +  (6)  =  log  [a  sin  |(^  _  7)]  =  8.44036 


6  +  c  =  0.25779  (2) 

^0  =  21°  7'. 3  (3) 

log  (6  -  c)  =  8.47056  (4) 

logcos|a  =  9.96980  (6) 

log  (6 +  c)=  9.41126  (5) 

log  sin  \  a  =  9.55673  (7) 


log  sin  ^  (/3  -  7)  =  9.45404 

log  cos  llp-y)=  9.98167 

(5)  +  (7)  =  log  [a  cos  ^  (^  _  7)]  =  8.96799 


(8) 
(12) 
(12) 

(9) 


(8 


log  tan  ^(/3- 7)  =9.47237 

(10) 

K/3-7)=10°31'.7 

(11) 

i(/3  +  7)  =  68"52'.7 

(14) 

/3  =  85°24'.4 

(15) 

7  =  52°21'.0 

(16) 

=  (9)  -  (12)  =  log  a  =  8.98632 

(13^ 

a  =  0.096900 

(17) 

2.  Solve  the  triangle  when  6  =  2.3671,  c  =  1.4345,  a  =  112°43'.4. 

Ans.  ^  =  42°  54'.5  ;  7  =  24°  22M  ;  a  =  3.2069. 

3.  Solve  the  triangle  when  a  =  101.47,  c  =  99.367,  /3  =  47°  48'..2. 

Ans.  a  =  67°  27'.1  ;  7  =  64°  44'.7  ;  6  =  81.396. 

102.    Case  IV.     Given  6,  c,  a.     Third  Method.  — To  find  the 


third  side  only. 


a^  =  h^  -\-  c^  —  2  be  cos  a. 
But 

cos  a  =  1  —  2  sin^  J-  a. 
...  a^  =  I?-{-(^-2bc  +  4bcsin^^a 
=  (6  -  (?)2  +  4  be  sin2  1  a 
NoT-      ibcsin^i'O] 

=^^-4^+-7i^)-H' 


4  ^(?  sin^  ^  a 


c*  -  ")■ 


OBLIQUE  PLANE   TRIANGLES.  107 

Let  X  be  an  angle  such  that 

^     „         4  ^<?  sin^  \  a 
tan^  X  =  -— -, r|— ; 

2siii^a    ^ 

or  tana5=— - — =— v6c.  (1) 

b  —  c  ^  ^ 

This  assumption  is  possible,  since  the  value  of  the  second 
member  of  (1)  must  lie  between  +qo  and  -co,  so  that  there 
will  always  be  some  angle  whose  tangent  is  equal  to  this 
quantity. 

.-.   a=(b  —  c^Vl -\- t'dii^  X  =  {h  —  c^secx; 

or  a=^-^.  (2) 

cos  05  ^    ^ 

First  find  x  from  (1),  and  then  a  from  (2).  In  these  equa- 
tions b  —  c  is  replaced  hy  c  —  b  when  c>  b, 

1.   Find  a  when  c  =  1.4345,  b  =  2.3671,  and  o  =  112°43'.4. 

log  tan  x  =  I  (log  b  +  log  c)  +  log  2  +  log  sin  ^  a  +  col  (b  —  c\. 
log  a  =  log  (6  —  c)  —  log  cos  X. 


log  b  =  0.37422 
logc  =  0.15070 

log  (6 -c)  =  9.96970 
-logcosic  =  9.46361 

log  6c  =  0.53092 

log  a  =  0.50609 

log  a/6c  =  0.26546 

log  2  =  0.30103 

logsin|a  =  9.92041 

col  (&-c)=  0.03030 

log  tan  ic  =  0.51720 
X  =  73°  5'.6 

a  =  3.2069 

2.  Find  6  when  a  =  101.47,  c  =  99.367,  jS  =  47°48'.2. 

.-.  x  =  88°  31'.  17;  6=81.396. 

3.  Find  a  when  6  =  19.937,  c  =  62.475,  a  =  130°  9'.4. 

.-.  x  =  56°23'.7;  a  =  76.858. 

OBLIQUE   TRIANGLES   SOLVED   BY   RIGHT  TRIANGLES. 

103.  Case  I.  Given  a,  a,  y,  —  In  Figs.  63  and  64,  on  the 
next  page,  draw  DB  perpendicular  to  AC  Considering  the 
first  figure,  in  the  triangle  BDO  we  know  a  and  7,  and  we 
compute  DB  and  BQ\  then  in  the  triangle  BDA  we  know 
DB  and  a,  and  we  compute  AD  and  <?;   then  b  =  AD  -{-  DC, 


108 


PLANE   AND  ANALYTICAL   TRIGONOMETRY. 


completing  the  solution.  In  the  second  figure,  where  7  is 
obtuse,  we  know,  in  the  triangle  BDC^  a  and  DCB  =  180°  — 7, 
and  we  compute  DB  and  CD\  then  in  the  triangle  BDA  we 
know  DB  and  a,  and  we  compute  c  and  AD\  then  h  —  AD 
—  QD^  completing  the  solution. 


Fig.  63. 


1.  Solve  the  triangle  when  a  =  3.4356,  o  =  17°  43'.4,  7  =  60°  35'.7. 

.-.  j8  =  101° 40'.9  ;  X>C=  1.6868;  i)J5  =  2.9929  ;  ^D  =  9.3650  ; 

c  =  9.8315;  &  =  11.0518. 

2.  Solve  the  triangle  when  a  =  54.376,  7  =  103°  3'.2,  /3  =  40°  10'. 3. 

Ans.  o  =  36°  46'.5  ;  c  =  88.478  ;  h  =  58.592. 

3.  Solve  the  triangle  when  c  =  230.47,  a  =  21°  32'.2,  /3  =  36'^  24'.4. 

Ans.  7  =  122°  3'.4  ;  a  =  99.825  ;  h  =  161.3975. 

104.  Case  II.  Given  a,  c,  a. — In  the  right  triangle  ABB 
we  know  c  and  «,  and  we  compute  AD  and  DB;  then  in  the 
triangle  CBD  we  know  DB  and  a,  and  we  find  DQ  and  7; 
then 


6  =  ^2)  + DC;    /3  =  180°-(«  +  7); 
b'  =  AD-DC;  y  =  180°-7;  ^8' =  180° -(«  +  7'). 

Two  solutions  are  possible  only  when  a  is  acute  and  a  is  less 
than  <?  and  greater  than  DB, 


OBLIQUE   PLANE   TRIANGLES. 


109 


If  a  is  obtuse,  as   in   Fig.  66,  we  solve   first  the  triangle 
BAD,  then  the  triangle  BCD,  and  find  b  =  DC-  DA. 


1.  Solve  the  triangle  when  c  =  23.647,  a  =  14.135,  a  =  33°  17'.3. 

.  •.   AD  =  19.767  ;  DB  =  12.979  ;  7  =  66°  40'.0  ;  DC  =  5.5986  ; 

r  7=    66°40'.0;  j8  =80°  2'.7  ;  b  =25.3656; 
\7'  =  113°20'.0;  ^'  =  33°22'.7;  6' =  14.1684. 

2.  Solve  the  triangle  when  a  =  2.4741,  c  =  1.0003,  a  =  6t)°  14'.8. 

••  7  =  22°  12'.8  ;  /3  =  88°  32'.4  ;  AD  =  0.35445  ;  DC  =  2.2905  ;  b  =  2.64495. 

3.  Solve  the  triangle  when  a  =  10.473,  b  =  12.987,  a  =  44°  11 '.3. 
^  /3  =    59°48'.5  ;  y  =  76°   0'.2  ;  c  =  14.5793  ; 


^ns. 


\(j'  =  120°  11'.5  ;  7'  =  15°  37'.2  ; 


4.0455. 


4.    Solve  the  triangle  when  a  =  0.43477,  b  =  0.40031,  a  =  94°  17'.6. 

Ans.  $  =  66°  39'.6  ;  7  =  19°  2'.8  ;  c  =  0.142282. 

105.   Case  III.     Given  a,  &,  c.  —  In  Fig.  63, 

p^  =  c^-  AD^;  jt?2  =  a2  _  j)02^ 

...  (^-AD^=a'^-DC\ 

,'.  AD^-DC^  =  (^-a^. 

(c  +  g)  (g  -  a)  _  (g  +  a^  (c  -  a') 
•  •  ^^~^^=~~AD^fDC~-  l  ' 

from  which  AD  —DC  may  be  computed.     Then 

AD=i[b+CAD-DC)], 
and  DC  =  ilb-(AD-DC)^, 

If  either  J.i)  or  DC  is  negative,  it  is  exterior  to  the  tri- 
angle; that  is,  the  point  D  is  on  the  line  AC  produced. 

Having  found  AD  and  DC,  the  angles  are  found  from  ths 
right  triangles  DBA  and  DBC. 

1.   Solve  the  triangle  when  a  =  27.103,  b  =  16.432,  c  =  12.511. 
...  c-a  =  -14.592;  ^i>  -  i)C  =  - 35.178  ;  .41)  =  -9.373;  2)0  =  25.805. 
^n.«?.  a  =  138°  31'.2  ;  7  =  17°  48'.6 ;  )8  =  23°  40'.3. 
In  this  example  D  lies  to  the  left  of  A. 


110 


PLANE   AND  ANALYTICAL  TRIGONOMETRY. 


2.  Solve  the  triangle  when  a  =  32.456,  h  -  41.724,  c  =  53.987. 

.-.  AD-  DC  =  44.(507  ;  AD  =  43.1655  ,  DC  =  -  1.4415. 

Alls,  a  -  36°  54'.7  ;  7  =  92°  32'. 7  ;  &  -  50°  32'.6. 

3.  Solve  the  triangle  when  a  =  0.14679,  h  -  0.10433,  c  =  0.04796. 
.-.  ^2>  -  i>C  =  - 0.18448  ;  ^Z)  =  - 0.040075  ;  DC  =  + 0.144405. 

Ans.  a  =  146°  40'. 75  ;  7  =  10°  2l'.0  ;  )8  =  22°  58'.25. 

106.  Case  IV.  Given  6,  c,  a.  —  In  the  triangle  J.i>^,  know- 
ing c  and  «,  find  AT)  and  BB.  Then  in  the  triangle  BBC  we 
know  BB  and  BC  =h  —  AB,  so  that  we  can  compute  a  and  7. 


1.  Solve  the  triangle  when  b  =  1143.7,  c  =  1822.4,  a  =  15°6'.4. 

.-.  ylZ>=  1759.5;     Z>^  =  474.96;  DC  =  -615.8. 

Ans.  7  =  142°  21'.5  ;  a  =  777.68  ;  3  =  22°  32'.  1. 
The  negative  value  of  DC  shows  that  D  is  to  the  right  of  C. 

2.  Solve  the  triangle  when  b  =  19.937,  c  =  62.475,  a  =  130°  9'.4. 

.-.  ^D  =  - 40.288;  DC  =  60.225. 

Ans.  7  =  38°  24'.5  ;  i8  =  11°  26M  ;  a  =  73.857,  or  76.858. 
Note  that  a  is  obtuse. 
8.    Solve  the  triangle  when  a  =  101.47,  c  =  99.367,  j8  =  47°48'.2. 

Ans.  7  =  64°  44'.6  ;  a  =  67°  27'.2  :  b  =  81.394. 


AREAS   OF  TRIANGLES. 

107.    Given  Two  Sides  and  the  Included  Angle  (6,  c,  a).— 
Represent  the  area  by  A.     From  geometry,  in  Fig.  67, 

A  =  \pb. 
But  p  =  c  sin  a. 

.'.  A  =  ^  6c  sin  a,  (1) 

or,  the  area  of  a  triangle  is  equal  to  half  the  product  of  the  two 
sides  multiplied  by  the  sine  of  their  included  angle. 


OBLIQUE  PLANE  TRIANGLES.  HI 

108.  Given  One  Side  and  the  Three  Angles  («»,  a,  p,  -y). — 
Substitute  in  (1),  Art.  107,  the  value  of  c  found  from  the  sine 
proportion,  _  ^  sin  7 

sin  P 

giving  ^=f-^-^w|P-  (1) 

109.  Given  the  Three  Sides  («,  h,  c).— We  have 

A  =  \hc  mna  =  hc  sin  J  a  cos  J  a. 
From  (2)  and  (3),  Art.  99,  we  have 

110.  Given  Two  Sides  and  the  Angle  Opposite  One  of  them 
(6,  c,  P). — First  find  7  by  the  formula 


c    . 
sin  7  =  -  sin 
0 

^. 

Then                           a  =  180° 
and                                  A  =  ^bc 

sin  a. 

EXAMPLES. 

1.  Find  the  area  when  b  =  0.14367, 
c  =  0.11412,  0  =  42°  14'.6. 

2.   Find  the  area  when  a  =  3.4356, 
o=17°43'.4,  7  =  60°36'.7. 

log  6  =  9.15737 

log  c  =  9.05737 

log  sin  a  =  9.82755 

col  2  =  9.69897 

log  ^=7.74126 
A  =  0.0055114 

.-.  3  =  101°  40'.9. 

loga2  =  21oga=    1.07200 

col  2=    9.69897 

logsini8=    9.99091 

log  sin  y  =    9.94010 

col  sin  a  =    0.51652 

log^=    1.21850 
A  =  16.539 

3.   Find  the  area  when  a  =  0.0093146,  b  =  0.0176530,  c  =  0.0095768. 

2s  =  0.0365444  logs=    8.26179 

5=0.0182722  \og(s-a)=    7.95219 

s  -  a  =  0.0089576  log  (s-b)=    6.79183 

s- 6  =  0.0006192  log(s-c)=    7.93929 

s-c=  0.0086954  2)10.94510-20 

sum  =  0.0365444  ^°S^  ^    5.47255-10 

a  check.  ■                     ^=   Q-QQQQ29686 


112  PLANE   AND  ANALYTICAL   TRIGONOMETRY. 

4.    Find  the  area  when  a  =  9.4672,  c  =  14.433,  o  =  11°  14'.3. 

log  c  =  1. 15936  log  a  =  0.97622  log  a  =  0.97622 

log  sin  0  =  9.28979  logc=  1.15936  logc  =  1.15936 

col  a  =  9.02378  col  2  =  9.69897  col  2  =  9.69897 

log  sin  7  =  9^47293           log  sin  ^=  9.67899              log  sin /3^  =  9.02259 

7=    17°17M  log^=  1.51354  log  ^1' =  0.85714 

7'  =  162°  42'.9  A  =  32.624  A'  =  7.1968 

.-.  /3=  151°28'.6  ■ 
/3'  =      6°   2'.8 

Note  that  log  A  and  log  A'  can  be  found  by  adding  log  sin  p  and  log  sin  /3' 
respectively  to  log  a  +  log  c  -f  col  2,  a  shorter  method  than  that  given  in  this 
example. 

6.    Find  the  area  when  a  =  0.013456,  b  =  0.023678,  a  =  40°  31 '.4. 

Ans.  0.00010351. 

6.  Find  the  area  when  c  =  43.145,  o  =  40°  40'.3,  0  =  60°  30'.3.  Ans.  538.19. 

7.  Find  the  area  when  a  =  1.4142,  b  =  1.6735,  c  =  2.8533.       A7is.  0.83826. 

8.  Find  the  area  when  a  =  14.135,  c  =  23.647,  a  =  33°  17'.3. 

Ans.  164.61  or  91.948. 

111.  Illustrative  Examples.  —  The  hearing  of  a  line  is  the 
angle  it  makes  with  the  magnetic  meridian,  shown  by  the  mag- 
netic needle.  The  letter  indicating  whether  the  line  is  meas- 
ured north  or  south  of  the  point  of  beginning  is  written,  then 
the  number  of  degrees  and  minutes  in  the  angle,  and  then  the 
letter  indicating  whether  the  line  lies  to  the  east  or  to  the  west 
of  the  magnetic  meridian.  Thus,  if  the  bearing  of  the  line  AB 
is  S.  60°  W.,  the  line  is  measured  from  A  to  the  west  of  south 
by  an  angle  of  60°. 

The  distances  and  the  angles  given  in  the  examples  are 
horizontal  unless  otherwise  specified. 

1.  From  a  point  on  a  horizontal  plane  the 
angle  of  elevation  to  the  top  of  a  crag  is  40°  28'.6, 
and  4163.2  feet  farther  away  in  the  same  vertical 
plane  the  angle  is  28°  50 '.4.  Find  the  distances 
from  the  points  to  the  top  of  the  crag,  and  its 
height  above  the  horizontal  plane. 

BD  =  13399  feet ;  AD  =  9956.2  feet ;    CD  =  6463.0  feet  j 
BC  =  11737  feet :   AC  =  7573.2  feet. 


OBLTQW:   PLANK   TRTANOLES. 


113 


3.  A  tower  160.43  feet  high  is  situated  at  tlie  top  of  a  hill  (Fig.  C9) ;  600 
feet  down  the  hill  the  angle  between  the  surface  of  the  hill  and  a  Jine  to  the 
top  of  the  tower  is  8°  40'.4.  Find  the  distance  to  the  top  of  the  tower,  and  the 
inclination  of  the  ground  to  a  horizontal  plane. 

.-.  ^BC  =  136°59'.7;    ^C  :- 726.60  feet ;   DAB  =  46°  59^.7. 


'>B 


Fig.  70. 


3.  To  find  the  horizontal  distance  from  a  point  A  to  an  inaccessible  point 
B  (Fig.  70),  the  horizontal  distance  AC  and  the  angles  a  and  y  were  measured 
and  found  to  be  1042.3  feet,  72°9'.4,  and  14°  13'.7,  respectively. 

•  .-.  AB  =  256.69  feet ;    CB  =z  994.15  feet. 

4.  To  find  the  distance  between  two  points  A  and  B  not  visible  from  each 
other  (Fig.  71).  —  Select  a  third  pohit  C  from  which  A  and  B  are  visible,  and 
measure  the  distances  (7^  =  444.38  feet,  C5  =  222.76  feet,  and  the  angle 
^C^=17''  17'.6.  Ans.  AB  =  240.97  feet. 


--~i>B 


Fig.  71. 


Fig.  72. 


5.  To  find  the  distance  from  a  point  A  to  another  point  JB,  the  latter  being 
inaccessible  and  invisible  from  A  (Fig.  72).  — Select  two  points  C  and  D  so 
that  C,  J.,  and  D  shall  be  in  the  same  straight  line,  A  and  B  being  visible  both 
from  C  and  from  D.  From  measurement  it  is  found  that  CA  —  456.72  feet, 
AD  =  490.74  feet,  y  =  71° 22'. 7,  5  =  36°  19'.4. 

.-.  CB  =  589.10  feet ;   DB  =  942.475  feet ;   AB  =  619.51,  or  619.53  feet. 

CROCK.  TRIG.  8 


114 


PLANE   AND  ANALYTICAL   TRIGONOMETRY. 


6.  To  find  the  elevation  of  the  top  of  a  church  steeple  D  (Fig.  78)  above 
the  horizontal  plane  ACB,  and  the  distances  of  the  steeple  from  A  and  B.  — Let 
the  horizontal  distance  AB  =  435.53  feet,  the  horizontal  angles  a  =  140°  40'.2  and 
/3  =  10°  7'.6,  and  the  vertical  angles  7  =  32°45'.6  and  7'  =  10°  7'.3. 

.  •.  AC=  156.95  feet ;   BC  =  565.74  feet ;   CD  =  100.99,  or  101.00  feet. 

The  agreement  of  the  values  of  CD  is  a  check  upon  the  observed  angles  and 
upon  the  computations. 


7.  To  find  the  elevation  of  the  top  of  a  church  steeple  D  (Fig.  74)  above 
the  iYfo  points  A  and  B,  not  in  the  same  horizontal  plane,  the  inclined  distance 
from  A  to  B,  and  its  angle  of  inclination  5  to  a  horizontal  plane  being  measured, 
as  well  as  the  angles  a,  ^,  7,  and  7',  shown  in  the  preceding  example.  —  Let 
^5=134.70  feet,  5=3°2'.7,  a=43°14'.8,  /3=63°17'.5,  7  =  56°36'.6,  7'  =  62°17'.3. 

[First  find  the  horizontal  distance  AF  and  the  vertical  distance  FB  in  the 
right  triangle  AFB  ;  then  solve  the  horizontal  triangle  AFC;  and  then  find  CD 
and  FD  from  the  right  triangles  ACD  and  BFD  respectively.] 

.  •.  AF=  134.51  feet ;   FB  =  7.1553  feet ;   FC  =  BE  =  96.135  feet ; 

AC  =  125.34  feet ;    CD  =  190.17  feet ;  ED  =  183.02  feet. 
Check:  CD  =  FB  +  ED. 


Fig.  75. 


8.  To  find  the  distance  between 
two  inaccessible  points  A  and  B. — 
Select  two  points  C  and  D  from  which 
both  A  and  B  can  be  seen,  and  measure 

CD  =  456.82  feet,       a  =  30°  40'.6, 
)8  =r  40°  14'.8,  7=35°16'.4, 

5  =  56°47'.4. 
.-.  .42) =449. 09  feet;    ^C=274.41  feet ; 
jBJ9  =  398.66  feet ;    5C=616.66  feet ; 
AB  =  405.57,  or  405.58  feet. 


OBLIQUE  PLANE  TRIANGLES. 


115 


9.   To  find  the  distance  between  two  inaccessible  points  A  and 

being  visible  from  only  one  accessible  point 

C.  —  Select  a  point  D  from  which  A  and  C 

are  visible,  and  another  point  E  from  which 

B  and  C  are  visible.     From   measurement 

CZ>=943.37  feet,        C^=673.33  feet, 

a  =  72°9'.3,  /3  =  60°17'.9, 

7  =  32°  14'.6,  8  =  67°  33'.9, 

e  =  19°  W.I. 

. '.  CA  =  1217.0  feet ;      CB  =  222.28  feet ; 

AB  =  1035.8  feet. 


B,  both 


10.  To  find  the  distance  between  two  inaccessible  points  A  and  B^  there 
being  no  accessible  point  from  which  both  A  and  B  are  visible  (Fig.  77).  — 
Select  the  points  C,  2>,  E,  and  F  so  that  A,  (7,  and  E  shall  be  visible  from  i>, 
and  D,  F,  and  B  from  E.  Measure  the  angles  a,  /3,  y,  5,  e,  and  d,  and  the  dis- 
tances CD,  DE,  and  EF,  Show  how  AB  may  be  found  from  the  data  thus 
obtained. 


Ar, 


Fig.  77. 


Fig.  78. 


11.  Two  points  A  and  B,  8763.6  feet  apart  (Fig.  78),  are  situated  at  the 
sea  level  in  the  same  north  and  south  line  ;  a  vessel  is  seen  at  C,  and  an 
hour  later  at  D.  The  required  quantities  are  AC,  BC,  AD,  BD,  CD,  and 
the  angle  that  CD  makes  with  the  north  and  south  line,  having  measured 
BAC=:  120°  30'.6,  BAD  =  30°  14'.4,  ABC  =  40°  18'.8,  ABD  =  140° 28'.2. 

.-.  AC=  17260  feet ;   BC  =  22985  feet ;  AD  =  34552.5  feet ;  BD  =  27340  feet ; 

ACD  =  63°  14'.5  ;  ADC  =  26°  29'.3  ;  BCD  =  44°  3'.8  ;        BDC  =  35°  46'.8  ; 

CD=SSed6,  38697,  or  38699  feet ; 

e=S60°-BAC-ACB-BCD=nG°  15'.0, 
or  =ABD+BDA-{-ADC=176°U'.9. 

12.  In  measuring  the  line  from  A  to  B,  whose  direction  was  known,  it  was 
necessary  to  pass  an  obstacle  at  F  (Fig.  79).  A  distance  CD  =144.31  feet  was 
measured,  making  an  angle  y  =  19°  53'.4  with  AB,  and  the  angle  5  =  140°  10'. 3 


116 


PLANE   AND   ANALYTICAL   TRIGONOMETRY. 


was  laid  off  with  the  transit.     It  is  required  to  find  the  distance  DE  to  the  line, 
the  distance  CE,  and  the  angle  e,  in  order  that  tlie  line  AC  may  be  prolonged. 

Arts.   CE  =  27L06  feet ;   DE  =  143.98  feet ;   e  =  160°  3'.7. 


x:;;'^a 


G        B 


Fig.  80. 


13.  Ill  passing  an  obstacle  at  F  it  was  necessary  to  use  the  broken  line 
CDEG  (Fig.  80).  The  distances  CD  and  DE  and  the  angles  7,  5,  and  e  were 
measured.  It  is  required  to  find  the  distance  EG  to  the  line  AB,  the  dis- 
tance CG,  and  the  angle  6,  when  CD  =  100.37  feet,  DE  =  94.367  feet,  7  =  80°, 

5  =  101°19'.8,  ande  =  110°. 
.-.  DCE=S7°[>3'.3;       DEC  =  iO°  AG'.O  ; 
CE  =  150.67  feet ;      EG  =  108.46  feet ; 
CG' =  151.22  feet ;  0  =  111°19'.8. 

14.  From  the  top  of  a  lighthouse  ABy 
200  feet  above  the  sea  level,  the  angle  of 
depression  to  a  ship  was  7  =  10°  14'.3  :  an  hour 
later  it  was  7'  =  11°  lO'.O;  the  horizontal  angle 
between  the  directions  of  the  ship  at  the  two 
instants  was  a  =  127°  14'.4.  Find  the  distance 
sailed  by  the  ship. 

.-.  AC  =  1107.3  feet ;         AD  =  1012.2  feet ; 
CD  =  1899.3  feet. 

15.  A  ladder  52  feet  long  is  set  20  feet  in 
front  of  an  inclined  buttress,'  and  reaches  46 
feet  up  its  face.  Find  the  inclination  of  the 
face  of  the  buttress. 

Ans.  ABC  =  do°51'.S,  or  95°51'.9. 


t^ 


^  16.    The  sides  of  a  city  block  measured  AB  =  423.24,  BC= 

162.36,  CD  =  420.81,  and  DA  =  160.62  feet,  the  first  two  sides 
being  perpendicular  to  each  other.  Find  the  angles  between 
the  other  sides. 

.-.  ^0-^453.31  feet;  BCA  =  C0°   0'.8; 

^^C  =  20°59'.2;  ACD  =  20°  ^b'.O  ; 

CAD  =  G8°    8'.8';  CZ>yl  =  91°    6'.4  ; 

BCD  =  89°  45'.S;  BAD  =  S9°   8'.0. 


Fig.  83. 


OBLIQUE  PLANE   TRIANGLES. 


117 


17.  A  ship  B  is  12  miles  S.  45°  W.  of  a  lighthouse  A, 
and  sails  S.  50°  E.  to  C,  a  distance  of  15  miles.  Find  its 
distance  from  the  lighthouse.  B 

Ans.  AC=  18.374,  or  18.375  miles.       ''^" 


Fig.  84. 

18.  In  surveying  a  field  a  thick  wood  prevents  the  measurement  of  the 
angle  ABD  and  of  the  distance  BD.  The 
angle  ABC  =  70°  14'.6  is  measured,  a  line 
^O  is  run  743.86  feet,  the  angle  BCD  is 
found  to  be  62°  14'.4,  and  the  distance  CD 
to  be  912.82  feet. 

.-.   CjB2)  =  68°28'.1;    CDB  =  i9°lT.5; 

BD  =  868.34,  868.30,  or  868.38  feet ; 
^i?Z)z=138°42'.7. 


Fig.  85. 


19.  The  distance  OE  and  its  bearing  E'OE  are  required,  the  engineer 
having  measured  the  distances  a,  b,  c,  d,  and  e  and  their  respective  bearings, 
N.  30°  W.,  S.  60°  E.,  N.  20°  E.,  N.  40°  W.,  and 
N.  50°  E. 

OE'  =  OA'  -  B'A'  JrB'C'+C>D>-{-  D'E' 
=  a  cos  30°  -  h  cos  60°  +  c  cos  20° 
+  d  cos  40°  -f  e  cos  50°. 
E'E  =  -AA'-\-  B"B  +  C"C-DD"^E"E 
=  -  a  sin  30°  +  b  sin  60°  +  c  sin  20° 
-  d  sin  40°  +  e  sin  50°. 
Then        OE  cos  E'OE  =  OE', 
OEsmE'OE  =  E'E; 

whence  OE  and  E'OE  can  be  found.    Then 

the  quadrant  of  E'OE  fixes  the  direction  of 

the  line  OE ;  thus,  if  E'OE  =  40°,  the  bearing  is  N.  40°  E.  ;   if  ^^0^"=  110°, 

the  bearing  is  S.  70°  E.  ;  if  E'OE  =  230°,  the  bearing  is  S.  50°  W. ;  if  E'OE  = 

310°,  the  bearing  is  N.  50°  W. 

20.  At  a  certain  point  the  angles  of  elevation  of  the  base  of  a  vertical 
tower  and  of  its  top  are  a  and  /3  respectively,  the  height  of  the  tower  being 
h  feet.     Prove  that  the  horizontal  distance  from  the  point  to  the  tower  is 


Fig.  86. 


118 


PLANE   AND  ANALYTICAL   TRIGONOMETRY. 


j^cosa  cos/3  cosec(i3  —  a),  and  that  the  elevation  of    its  top  above  the  point 
is  h  cos  a  sin  /3  cosec  (/3  —  a) . 

21.  At  the  top  of  a  vertical  tower  whose  height  is  h,  the  angles  of  depression 
to  two  points  M  and  N  in  the  same  vertical  plane  with  the  tower  were  o  and  /3 
respectively  (/3  >  a),  the  points  being  in  the  same  horizontal  plane  with  the  base 
of  the  tower.     Prove  that  the  distance  MN  is  h  sin  (/3  —  a)  cosec  o  cosec  /3. 

22.  Two  points  M  and  iV  in  a  horizontal  plane  are  in  the  same  vertical  plane 

with  a  tower.  The  angle  of  elevation  of  the 
top  of  the  tower  from  31  is  a,  and  from  N  it' 
is  iS,  /3  being  greater  than  a.  Prove  that  the 
horizontal  distance  of  the  tower  from  iV"  is 
MN  sin  a  cosjS  cosec  {^  —  a). 


23.  Three  points,  A,  B,  and  C,  are  in  the 
same  horizontal  line,  the  distances  AB  and 
BG  being  a  and  b  feet  respectively  (Fig.  87). 
The  angles  of  elevation  of  the  top  of  a  tower 
measured  at  A,  B,  and  C  were  a,  /3,  and  y 
respectively.  Find  the  elevation  of  the  top 
of  the  tower  above  the  horizontal  plane 
through  the  points,  and  the  horizontal  dis- 
tances of  the  tower  from  the  three  points. 


Fig.  87. 


m 

=  h cot  a  ; 

n  =  h  cot ^;  p  = 

/i  cot  7  ; 

m^ 

=  a^-\-n^ 

-2  an  cos  ABD  ; 

P' 

=  b-2  +  W2 

+  2bn  cos  ABD ; 

a2 

+  n2- 

-m2 

p^-  62 

-  w2 

2  an 

2bn 

> 

. 

h^ 

ab(a  +  b) 

a  (cot2  7  -  cot2  /3)  +  6  (cot2  a  -  cot2  /3) 

24.  In  Fig.  88  the  distances  a  and  &  and  the  angles  a,  /3,  and  y  are  known, 
and  the  distance  BG  =  x  is  required,  ABGD 
being  an  inaccessible  straight  line. 


^<^r;||v: 


FB 


a  +  X 


FG 


Fig.  88. 


sin  a      sin  J. ' 

sin  (a  +  jS)      sin  A ' 

FB         a 

sin  (a  +  /S) 

FG     a  +  x 

sin  a 

b          FG  . 

b  +  x       _  FB 

sin  7      sin  Z) ' 

sin  (^  +  7)      sin  D 

FG         b 

sin(j8  +  7). 

FB     6  +  x 

sin  7 

Multiplying  (1) 

and  (2),  we  have 

(a  +  x)  (6  +  x)  sin  a  sin  7  =  ab  sin  (a  +  jS)  sin  (^  +  7), 
from  which  x  may  be  found,  since  the  equation  is  a  quadratic  in  x. 


(1) 


(2) 


OBLIQUE  PLANE   TRIANGLES. 


119 


a  tower 


25.  Two  points  A  and  B  in  the  same  vertical  plane  with  the  top  of 
are  on  a  sidehill  whose  angle  of  inclination  to  a 
horizontal  plane  is  5,  the  inclined  distance  AB 
being  a  feet.  The  angles  of  elevation  of  the  top 
of  the  tower  were  measured  at  A  and  B,  and 
found  to  be  a  and  )3.  Prove  that  the  horizontal 
distance  of  the  top  of  the  tower  from  B  is 

a  (cos  S  tan  a  —  sin  5)  cos  a  cos  0  cosec  (3  —  a), 

and  that  the  elevation  of  the  top  above  B  is 

a  (cos  5  tan  a  —  sin  5)  cos  a  sin  |3  cosec  (/3  —  a). 

26.  In  a  hydrographical  survey,  the  distances  between  three  points,  A,  Bj 
and  C,  on  the  shore  having  been  determined,  the  observer  in  the  boat  P  measures 
the  angles  5  and  e  subtended  by  AB  and  BC.  It  is  required  to  find  the  dis- 
tances of  the  boat  from  the  three  points. 

(1)  Graphical  Solution.  — Construct  on  AB  the  segment  of  a  circle  APB 
that  shall  contain  the  measured  angle  5,  and 
on  BC  the  segment  of  a  circle  BPC  that 
shall  contain  the  angle  e.  Their  point  of  in- 
tersection P  will  be  the  position  of  the  boat. 
There  are  four  possible  solutions,  only  one 
being  shown  in  the  figure.  .' 

(2)  Analytical  Solution. — Let  J.Z>CP  » 
be  the  circle  through  A,  C,  and  P.  Then 
DAC=  e,  and  DC  A  =  8.  Hence  in  the  tri- 
angle ADC  we  know  one  side  AC  and  the 
three  angles  ;  find  AD  and  CD.  In  the  tri- 
angle ABC  we  know  the  three  sides;  find  ^ 
the  three  angles.     In  the  triangle  DAB  we                              '^^^-  ^^^ 

know  two  sides  and  the  angle  DAB  =  CAB  —  CAD  ;  find  ABD.  Then  in  the 
triangle  ABP  we  know  one  side  and  the  three  angles  ;  find  AP  and  BP.  Also, 
compute  DBC  from  the  triangle  DCB,  and  then  BP  and  CP  from  the  triangle 
BPC.     The  values  of  BP  should  agree. 


In  the  following  examples  find  the  last  three  elements,  the  first  three  being 
given: 

27.  a  =  1.0431,      /3  =  4°4'.4,        7  =  22°3'.6. 

.'.  a  =  153°  62'.0  ;  b  =  0.16822  ;     c  =  0.88942. 

28.  a  =  103.37,       a  =  10°  11 '..3,     /3  =  83°43'.6. 


7-. 


M:       &  =  580.89:       c  =  583.02. 


29.  c  =  74.344,      a  =  105°  6'. 7,     /S  =  60°  14'.4. 

.-.  7  =  14°38'.9;     a  =  283.82  ;       6  =  255.21. 

30.  c  =  0.047365,  p  =  40°  7'. 7        y  =  39°  41'. 9. 

.'.  a  =  100°  10'.4  ;  a  =  0.072990  ;  b  =  0.047792. 

31.  c  =  4.4479,       a  =  11°  11'.3,     y  =  57°  37'.4. 

.  •.  /3  =  111°  11'.3  ;  a  =  1.0219 ;      b  =  4.9106. 


120  PLANE   AND   ANALYTICAL  TRIGONOMETRY. 

82.  b  =  143.97,   /3  =  30° 36'.8,  y  =  107°  15'.5.    .•'. 

.-.  a  =42°7'.7;   a  =  189.64;     c  =  269.98. 

33.6  =  10.467,   c  =  1.4321,   /S  =  114°  10'.3. 

.-.  7  =  7°10'.2;   a  =  58°39'.5;   a  =  9.79875. 

34.  a  =  0.67375,  b  =  0.43213,  a  =  147°  11'.3. 

.-.  i3  =  20°20'.2;   7  =  12°28'.5;   c  =  0.26858. 

35.  a  =  1.4742,   c  =  0.97674,  o  =  25°  19'.9. 

.  •.  7  =  16°  28'.  1  ;  j3  =  138°  12'.  0  ;  6  =  2. 2966. 

36.  a  =  943.42,  b  =  647.15,  a  =  104°  6'.9, 

.-.  /3  =41°42'.0;   7  =  34°  11'.  1;   c  =  646.59. 

37.  a  =  0.10321,  c  =  0.047323,  a  =  45°9'.7. 

.-.  7  =  18°58'.4;   /3  =  115°51'.9;  6  =  0.13097. 

38.  a  =  4.4321,   c  =  5.4763,   7=100°11'.9. 

'  .-.  o  =52°48'.l;  /3  =  27°0'.0;    6=2.5261. 

39.0=23.111,   6  =  19.476,   7  =  47°16'.7. 

.-.  ^=38°15'.0;   a  =  94°28'.3;   a  =  31.363. 

40.  a  =  0.11111,  0  =  0.12767,   a  =  23°  15'.6. 

.  •.  7  =  26°  59'.  1  ;  /3  =  129°  45'. 3  ;  6  =  0.21630  ; 
7'  =  153°  0'.9  ;  /3'  =  3°  43'.5  ;  6'  =  0.018279. 

4L  6  =  1.4326,   c=  1.3671,   7  =  44°  17'.3. 

.-.  /3=  47°  1'.9;  a  =  88°40'.8;   a  =  1.9574  ; 
/3'  =  132°  58'.  1 ;  a'  =  2°  44'. 6  :  a'  =  0.093706. 

42.  a  =  46.703,   6  =  57.147,   a  =  19°  17'. 7. 

.-.  /3  =  23°50'.9;  7  =  136°51'.4;  c  =  96.652  ; 
/3'  =  156°  9'.1  ;  7'  =   4°33'.2  ;  c'  =  11.221. 

43.  a  =  9.4327,   c  =  10.4751,  a  =  63°  17'.3. 

.-.  7  =82°45'.0;   ^  =  33°57'.7;   6  =  5.8990; 
7'  =  97°  15'.0  ;  ^'  =  19°  27'.7  ;   6'  =  3.5182. 

44.  a  =  0.034337,  c  =  0.062774,  a=9°6'.7. 

.:  y=    16°  49'. 7  ;  ^  =  154°  3'.6  ;  6  =  0.094846  ; 
7'  =  163°10'.3;  /3' =  7°43'.0;  6' =  0.029115. 

46.  a  =  0.79797,  6  =  0.46731,  jS  =  23°  19'.6. 

.-.  a=  42°32'.5;  7  =  114°  7'.9 ;  r.=  1.07705; 
a'  =  137°27'.5;  7'=  19°  12'.9  ;  c' =  0.38841. 

46.  a  =  37.456,   6  =  43.987,   c  =  13.498. 

.  •.  i  a  =  26°  31'.0  ;  ^  jS  =  55°  7'.0  ;   i  7  =  8°  22'.0. 

47.  a  =  2.4568,   6  =  2.4743,   c  =  1.0047. 

.-.  ia  =  38°38'.0;  i^  =  39°36'.7;  i7  =  ll°45'.3. 

48.  a  =  47.474,   6=100.980,   c  =  93.929. 

.-.  ia  =  13°56'.8;  |^  =  42°10'.2;  i7  =  33°53'.0. 

49.  a  =14.567,   6  =  9.4769,   c  =  11.113. 

.-.  ia  =  44°50'.9;  i /3  =  20°  17'.5  ;  i7  =  24°51'.5. 


OBLIQUE   PLANE  TRIANGLES.  121 

50.   rt  =  2.1476,  6  =  1.9397,      c  =  3.4345. 

.-.  ^0=17^22^8;    i/3=15°29'.8;    ^y  =  b7°T.3. 

61.  a  =  115.03,  b  =  129.15,     c  =  112.06. 

.-.  ia  =  28°12'.9;   J /3  =  34°  39'.2  ;    J  7  =  27°  7'.9. 

52.  b  =  113.47,  c  =  227.79,     a  =  19°  43'.4. 

.-.  /3=  17°33'.8;      7  =  142°42'.8;     a  =  126.90  ; 
or  log  tan  x  =  9.68278 ;        a  =  126.89. 

53.  a  =  99.416,  c  =  90.432,     /3  =  ll°7'.8. 

.-.  a  =  110°20'.4;    7  =  58°31'.8;       6  =  20.467; 
or  log  tan  x  =  0.31110  ;        b  =  20.467. 

54.  a  =  1.4342,  b  =  9.7672  ;   7  =  109°  19'. 0. 

.-.  o=7°31'.7;        ^  =  63°8'.7;         c  =  10.330,  or  10.331 ; 
or  log  tan  x  =  9. 86498  ;        c  =  10.331. 

55.  a  =  1003.7,  b  =  943.67,     7  =  101°  19'.8. 

.-.  a  =  40°46'.9;      i8  =  37°53'.3;       c  =  1506.7  ; 
or  log  tan  x  =  1.39930  ;        c  =  1506.7. 

56.  a  =  222.76,  6  =  444.38,     7  =  17°  17'.6. 

.-.  a  =  15°57'.0;      /S  =  146°45'.4  ;     c  =  240.97; 
or  log  tan  x  =  9.63029  ;        c  =  240.97. 

67.  a  =  363.24,  6  =  146.18,     7  =  68°  14'.4. 

.-.  a  =  88°2'.6  ;        /3  =  23°43'.0  ;       c  =  337.55,  or  337.56; 
or  log  tan  x  =  0.07590  ;        c  =  337.55. 


PART   TWO. 

SPHERICAL  TBIGOJfOMETBY. 

^-if^Oo 

CHAPTER   YIII. 
DEFINITIONS  AND  CONSTRUCTIONS. 

112.  Spherical  Trigonometry  treats  of  the  relations  between 
the  face  angles  and  the  edge  angles  of  a  trihedral  angle. 

An  edge  angle  is  the  angle  between 

two  of  the  three  planes  forming  the  ^--^t^'^    rV 

trihedral   angle  ;    it   is   measured    by  ct<<^^^^^ --^^^^^    ?l   A 

the  angle  between  the  lines  cut  from  >^>i5^^^        \    <      j  ry 

the   two   planes    by   a    plane    perpen-  ^^^^^'^^'    J^ 

dicular  to  the  edge  in  which  the  two  b^^^-J^ 
planes  intersect. 

A  face  angle  is  the  angle  between  two  of  the  edges. 

113.  Representation  of  Trihedral  Angles.  —  The  relations 
between  the  elements  of  a  trihedral  angle  are  discussed  by 
means  of  the  spherical  triangle  formed  by  the  intersections  of 
the  faces  with  a  sphere  described  with  any  radius  about  the 
vertex  as  a  center.  The  faces  will  cut  arcs  of  great  circles 
from  the  surface  of  the  sphere,  their  angular  measures  being 
the  same  as  those  of  the  face  angles;  and  the  angles  of  the 
spherical  triangle  will  correspond  to  the  edge  angles,  each 
being  measured  by  the  angle  between  two  lines  lying  in  the 
planes  of  the  faces  and  perpendicular  to  the  line  of  intersection 
of  the  faces. 

1^ 


124  SPHERICAL   TRIGONOMETRY. 

Hence,  in  the  spherical  triangle  the  sides  correspond  to  the 
face  angles,  and  the  angles  to  the  edge  angles  of  the  trihedral 
angle. 

The  lengths  of  the  sides  in  linear  measure  will  depend  upon 
the  radius  of  the  sphere,  and  are  computed,  when  the  radius  is 
known,  by  the  proportion 

360°:a  =  2Trr:«,  (1) 

where  a  is  the  number  of  degrees  in  the  arc,  and  I  is  its  length. 

114.  Limitation  of  Values.  —  We  shall  consider  only  those 
triangles  in  which  each  element  is  less  than  180°.  In  the  gen- 
eral spherical  triangle  the  sides  and  angles  may  have  values 
greater  than  180°,  but  in  such  a  case  it  is  always  possible  to 
substitute  for  the  triangle,  in  the  computations,  another  in 
which  each  element  shall  be  less  than  180°. 

115.  Definitions  and  Relations. — A  great  circle  is  cut  from 
the  surface  of  a  sphere  by  a  plane  passing  through  its  center; 
its  radius  is  equal  to  the  radius  of  the  sphere. 

A  %mall  circle  is  cat  from  the  surface  by  a  plane  not  passing 
through  the  center;  its  radius  is  always  less  than  the  radius  of 
the  sphere. 

Two  planes  passing  through  the  center  will  intersect  in  a 
diameter  of  the  sphere,  and  the  two  corresponding  great  circles 

will  intersect  at  the  ends  of  this  diame- 
ter. Hence  any  two  great  circles  will 
intersect  at  two  points  180°  apart. 

To  describe  a  great  circle  on  a 
sphere,  separate  the  points  of  a  pair 
of  compasses  by  a  distance  equal  to 
the  chord  of  90°,  or  rV2,  and  describe 
an  arc  about  any  point.  If  any  other 
^     _  distance  is  used,  a  small  circle  will  be 

Fig.  92. 

described.  The  point  used  as  the 
center  is  called  the  fole  of  the  great  circle;  its  distance  from 
all  points  on  the  great  circle  is  evidently  90°. 

Any  great  circle  passing  through  the  pole  of  another  great 
circle  will  be  perpendicular  to  that  great  circle 


DEFINITIONS   AND  CONSTRUCTIONS.  125 

Any  two  great  circles  drawn  perpendicular  to  a  third  great 
circle  will  intersect  in  its  pole. 

A  great  circle  perpendicular  to  two  great  circles  will  pass 
through  the  poles  of  both,  and  its  plane  will  be  perpendicular 
to  the  diameter  joining  the  points  of  intersection  of  the  two 
great  circles. 

The  angle  between  two  arcs  of  great  circles  is  measured 
by  the  arc  of  a  great  circle  described  about  the  vertex  as  a 
pole,  and  limited  by  the  sides,  produced  if  necessary 

The  shortest  distance  between  two  points  on  a  sphere  is  the 
arc  of  the  great  circle  passing  through  the  points. 

116.  Constructions.  —  To  find  the  pole  of  a  given  great  cir- 
cle: from  any  two  points  on  the  circle  as  poles,  describe  arcs  of 
great  circles,  and  their  intersection  will  be  the  point  required. 

To  draw  a  great  circle  through  two  points:  find  the  pole  as 
before,  and  describe  the  great  circle. 

To  draw  a  great  circle  through  a  given  point  perpendicular 
to  a  given  great  circle:  from  the  point  as  a  pole  describe  an 
arc  of  a  great  circle;  its  point  of  intersection  with  the  given 
circle  will  be  the  pole  of  the  required  circle.  Or,  find  the  pole 
of  the  given  circle,  and  then  draw  the  great  circle  through  this 
pole  and  the  given  point. 

To  cut  from  a  great  circle  an  arc  n°  long:  separate  the 
points  of  the  compasses  by  a  distance  equal  to  the  chord  of 
n^^  or  2  r  sin  ^  7i°,  place  the  points  on  the  great  circle,  and  the 
arc  intercepted  will  be  the  one  re- 
quired. 

To  construct  a  great  circle  pass- 
ing through  a  given  point  and  mak- 
ing a  given  angle  with  a  given  great 
circle:  in  Fig.  93,  let  ACB  be  the 
given  great  circle,*  P  its  pole,  F  the 
given  point,  and  a  the  given  angle. 
With  P  as  a  pole,  draw  the  small 
circle  P'P"   such  that  the  angular  FilTaT 

*  The  planes  of  the  great  circles  ACB  and  CF,  and  of  the  small  circle 
P'P"^  are  perpendicular  to  the  paper. 


126  SPHERICAL   TRIGONOMETRY. 

distance  PP'  =  a;  then  the  pole  of  the  required  great  circle 
must  be  on  this  small  circle.  With  jP  as  a  pole,  describe  an 
arc  of  a  great  circle  cutting  the  small  circle  P'P"  in  two  points; 
these  points  will  be  the  poles  of  two  great  circles  through  F, 
both  of  which  satisfy  the  given  conditions.  Only  the  great 
circle  OF,  whose  pole  is  P',  is  shown  in  the  figure. 

To  construct  a  great  circle  making  a  given  angle  with  a 
given  great  circle,  the  point  of  intersection  being  given:  from 
the  given  point  as  a  pole  describe  a  great  circle,  lay  off  on  it 
from  the  given  circle  a  distance  equal  in  angular  measure  to 
the  given  angle,  and  pass  a  great  circle  through  the  point  thus 
found  and  the  given  point  of  intersection. 

117.  Definitions. — A  right  spherical  triangle  is  one  which 
has  one  angle  equal  to  90°;  a  birectangular  triangle  lias  two 
angles  each  equal  to  90°;  a  trirectangular  triangle  has  three 
angles  each  equal  to  90°. 

A  quadrantal  triangle  has  one  side  equal  to  a  quadrant,  or 
90°;  a  biquadrantal  triangle  has  two  sides  each  equal  to  a 
quadrant;  a  triquadrantal  triangle  has  three  sides  each  equal 
to  a  quadrant. 

A  birectangular  triangle  is  also  biquadrantal,  and  a  tri- 
rectangular triangle  is  also  triquadrantal;  and  vice  versd. 

118.  The  Polar  Triangle  of  any  triangle  is  constructed  by 
describing  arcs  of  great  cireles  about  the  vertices  of  the  origi- 
nal triangle  as  poles.  Thus,  about  A, 
B,  apd  C  as  poles,  describe  the  arcs 
B'  C\  A!'Q\  and  A! B\  respectively;  that 
triangle  is  called  the  polar  in  which 
the  vertices.  ^  and  A! ,  B  and  B\  C  and 
C  are  on  th'^  same  side  of  BO,  AO,  and 
AB,  respectively. 

The  vertices  of  the  polar  triangle 
will  be  the  poles  of  the  sides  of  the 
original  triangle,  so  that  either  triangle  will  be  the  polar  of 
the  other. 

The  sides  of  a  triangle  are  the  supplements  of  the  opposite 
angles  of  the  polar,  and  the  angles  are  the  supplements  of  the 


DEFINITIONS   AND   CONSTRUCTIONS. 


127 


Fio.  95. 


opposite  sides  of  the  polar;  a'=  180°—  a,  aJ  —  180"—  a.     Thus, 
if  the  angles  of  a  triangle  be  120°, 
80°,  and  60°,  the   opposite   sides  of 
the  polar  will  be  60°,  100°,  and  120°. 

The  polar  of  a  quadrantal  tri- 
angle is  a  right  triangle,  the  angle 
in  the  polar  opposite  the  quadrant 
being  equal  to  the  supplement  of 
90°;  the  polar  of  a  biquadrantal  tri- 
angle is  birectangular;  the  polar  of 
a  triquadrantal  triangle  is  trirectan- 
gular;  and  vice  versd. 

The  triquadrantal  triangle  is  its  own  polar,  each  vertex 
being  the  pole  of  the  opposite  side. 

119.  In  Any  Spherical  Triangle  : 

(1)  Each  side  must  be  an  arc  of  a  great  circle. 

(2)  Each  side  must  be  less  than  the  sum  of  the  other  two. 

(3)  The  greater  side  is  opposite  the  greater  angle,  and 
conversely.     Equal  sides  are  opposite  equal  angles. 

(4)  The  sum  of  the  sides  must  be  less  than  360°. 

(5)  The  sum  of  the  angles  must  be  greater  than  180°  and 
(ess  than  540°. 

120.  Construction  of  Triangles.  —  (1)  Given  the  three  sides, 
a,  5,  c.  — Draw  an  arc  of  a  great  circle  and  lay  off  on  it  an  arc 
equal  to  one  of  the  sides,  as  a.  From  the  extremities  of  this 
arc  as  poles,  with  radii  equal  to  the  chords  of  h  and  c  respec- 
tively,, describe  arcs  of  small  circles  with  the  compasses,  and  find 
their  point  of  intersection.  Join  this  point  and  the  extremities 
of  a  by  arcs  of  great  circles,  and  the  triangle  will  be  constructed. 

(2)  Given  the  three  angles,  a,  /3,  7.  —  Find  the  sides  of  the 
polar  triangle,  construct  it,  and  then  construct  the  given  tri- 
angle by  using  the  vertices  of  the  polar  as  poles. 

(3)  Given  two  sides  and  the  included  angle,  a,  5,  7.  —  Draw 
an  arc  of  a  great  circle,  and  lay  off  on  it  an  arc  equal  to  one 
of  the  sides,  as  a.  Pass  an  arc  of  a  great  circle  through  one 
extremity  of  a,  making  the  angle  7  with  a,  and  lay  off  on  it  an 
arc  equal  to  h.  Join  the  extremities  of  a  and  h  by  an  arc  of  a 
great  circle,  and  the  triangle  will  be  constructed. 


v'^V^^' 


ouw^^^ 


s^ 


t>f 


CAi  \rcS 


128 


SPHERICAL   TRIGONOMETRY. 


(4)  Given  two  angles  and  their  included  side,  a,  /3,  o.  —  In 
the  polar  we  know  two  sides  and  the  included  angle,  and  hence 
we  can  construct  it  by  the  method  just  given.  Having  the 
polar,  we  can  then  construct  the  required  triangle. 

Or,  draw  a  great  circle  and  lay  off  on  it  an  arc  equal  to  e  ; 
at  the  extremities  of  this  arc,  construct  arcs  of  great  circles 
making  the  angles  a  and  y3  with  c  ;  their  point  of  intersection 
will  be  the  third  vertex. 

(5)  Given  two  sides  and  the  angie  opposite  one  of  them, 
a,  5,  a.  —  Draw  any  great  circle  ADA\  and  through  any  point 

on  it,  as  A^  draw  a  great  circle 
making  the  angle  DA  C=  a  with 
it.  On  this  circle  l:iy  off  from 
A  the  distance  AC  =h.  With 
-^       ^    ^     ^  ^6^  as    a   pole,   describe    a   small 

Fig.  96.  •       ^  ^  ^ 

Circle  whose  radius  is  equal  to 
the  chord  of  a,  using  the  compasses ;  pass  arcs  of  great  circles 
through  C^  and  the  points  B  and  B'  where  this  small  circle 
intersects  the  first  great  circle  ADA'^  and  the  triangle  will  be 
constructed. 

There  will  be,  in  general,  two  points  of  intersection,  and 
there  may  therefore  be  two  triangles  that  will  satisfy  the  con- 
ditions of  the  problem.  Only  those  triangles  can  be  taken  in 
which  each  side  is  less  than  180°,  i.e.  both  B  and  B'  must  lie 
on  the  arc  ADA'  between  A  and  A\  these  points  being  180° 
apart. 

If  a  is  acute,  as  in  Fig.  96,  a  must  be  greater  than  p  and 

less  than  the  shorter  of  the  two 
distances  CA  and  CA'  (h  and 
180°  -  6)  in  order  that  there 
may  be  two  solutions. 

If  a  is  obtuse,  as  in  Fig.  97, 
CD'  is  the  least  and  CD  the 
greatest  distance  of  C  from 
ADA'D',  DCD'  being  perpen- 
dicular to  ADA'D'.  Therefore 
a  must  be  less  than  jt?,  in  ordei 
that  the  small  circle  may  cui 
■piQ,  97.  ADA'D' ;  a  must  also  be  greatei 


DEFINITIONS   AND  CONSTRUCTIONS.  129 

than  the  longer  of  the  two  distances  CA  and  CA'  (6  and 
180°  —  6)  in  order  that  the  two  points  of  intersection  may  fall 
on  the  arc  ABA'. 

The  conditions,  therefore,  for  two  solutions  are  : 

a  acute  :     a  >  p^    a  <  b^   a  <  180°  —  b. 
a  obtuse  :  a  <p,    a>  b^   a>  180°  —  b. 

Or,  a  must  be  intermediate  in  value  between  p  and  both  b  and 
180°  -  b. 

If  a  is  intermediate  in  value  only  between  p  and  either  b  or 
180°  —  6,  there  will  be  one  solution. 

If  a  is  not  intermediate  in  value  between  p  and  either  b  or 
180°  —  6,  no  solution  will  be  possible,  but  ii  p  =  a,  there  will 
be  one  solution  —  a  right  triangle. 

(6)  Given -two  angles  and  the  side  opposite  one  of  them, 
«,  /3,  a.  —  In  the  polar  triangle  we  know  two  sides  and  the 
angle  opposite  one  of  them,  and  we  can  construct  it ;  having 
the  polar  we  can  construct  the  required  triangle. 

As  the  polar  triangle  may  admit  of  two  solutions,  there  may 
be  two  solutions  of  the  problem. 

CROCK.    TRIG.  —  9 


CHAPTER    IX. 


GENERAL   FORMULAS. 


Fia.  98. 


121.   The  Cosine  of  Any  Side  of  a  Spherical  Triangle  is  equal 
to  the  product  of  the  cosines  of  the  other  two  sides,  increased  hy  the 

product  of  the  sines  of  these  two  sides 
multiplied  hy  the  cosine  of  their  included 
angle. — Let  the  phxne  BAC  be  per- 
pendicular to  OA  at  any  point  A,  and 
let  J.  (7,  BC,  and  BA  be  its  intersections 
with  the  faces  of  the  trihedral  angle. 
Then  BA  C  —  a,  and  AB  and  A  0  are 
perpendicular  to  OA,  i.e.  OAB  and  OAO  are  triangles  right- 
angled  at  A. 

In  the  triangle  BA  0  we  have 

BC^  =  AB^  -{- AC^- 2  AB' AC  cos  a. 

In  the  triangle  BOO, 

BC'^=  0&-{-  OC^-I  OB'  00  cos  a. 

Equating  the  values  of  BC^,  and  transposing, 

2  OB  '  O0cosa  =  OB^  -  AB^  +  00^  -  AO^  +  2AB'  AOcosa. 

In  the  right  triangles  OAB  and  OAO, 

OB"-  -  AB"^  =  0A\  and   00^  -  AO^  =  OA^. 

.'.  2  OB'  00  cos  a  =  OA^  +  OA'^  +  2  AB  ■  AC  cos  a; 

OB  '  00  cos  a  =  OA^  +  AB  ■  AC  cos  a. 


or 


or 


cos  a 


cos  a 


OA 

00 


OA     AC 

OB      00 


AB 
OB 


cos  a 


cos  b  cos  c  +  sin  6  sin  c  cos  a. 

130 


(1) 


GENERAL   FORMULAS. 


131 


In  this  proof  b  and  c  are  assumed  to  be  less  than  90°,  while  a  and  a  may 
have  any  values  less  than  180°.  The  formula  is  true,  however,  when  either  b 
or  c,  or  both  b  and  c,  exceed  90°. 

If,  in  the  triangle  represented  by  the  full  lines  (Fig.  99),  b  is  greater  than 
90°,  then  in  the  dotted  triangle  formed  by  completing  the  arcs  of  great  circles, 
the  two  sides  are  180°  —  6,  and  c,  both  less  than  90°,  and  the  other  side  and  its 
opposite  angle  are  180°  —  a,  and  180°  —  a.  Hence  we  can  apply  (1)  to  the 
dotted  triangle,  giving 


Fkj.  100. 

cos  (180°  -  a)=  cos  (180°  -  b)  cose  +  sin  (180°  -  6)  sine  cos  (180°  -  o). 
.-.  —  cosa  =  —  cos6  cose  —  sin6  sinecoso. 

.*.  cos  a  =  cos  6  cose  +  sin  &  sine  cos  a.  q.e.d. 

If  both  b  and  c  are  greater  than  90°,  as  in  Fig.  100,  then  in  the  dotted 
triangle  the  two  sides  are  180°  —  b,  and  180°  —  c,  and  the  other  side  and  its 
opposite  angle  are  a  and  a. 

.-.  cos  a  =  cos  (180°  -  b)  cos  (180°  -  c)  +  sin  (180°  -  b)  sin  (180°  -  e)  cos  a 

=  (  —  cos  &)  (  —  cos  c)  +  sin  b  sin  e  cos  a. 

.-.  cos  a  =  cos  6  cos  c -f  sin  6  sin  c  cos  o.  q.e.d. 

Therefore  the  formula  is  always  true  when  each  of  the  elements  of  the 
triangle  is  less  than  180°. 

No  assumption,  then,  has  been  made  concerning  any  element 
that  is  not  true  for  all  the  others.  We  may  therefore  change 
any  angle  to  another,  as  a  to  yS,  if  at  the  same  time  we  change 
the  sides  opposite,  as  a  to  6,  making  also  the  reverse  changes, 
5  to  a  and  ^  to  a,  in  the  formula  ;  for  this  is  equivalent  to 
changing  the  names  arbitrarily  assigned  to  the  sides  and  angles. 
Thus,  to  permute  (1)  to  find  cos  <?,  we  change  a  to  <?,  «  to  7,  e  to 
a,  and  7  to  a,  if  they  occur  in  the  formula,  while  b  and  yS  will 
not  be  affected. 

.  •.  cos  c  =  cos  b  cos  a  H-  sin  5  sin  a  cos  7. 

If  we  assume  that  our  triangle  is  right-angled,  7  being 
equal  to  90°,  we  can  permute  between  a  and  6,  and  d  and  yS, 
since  no  assumption  is  made  concerning  a  and  a  that  is  not 
equally  true  concerning  b  and  y8.      But  we  cannot  permute 


132 


SPHERICAL    TRIGONOMETRY. 


between  a  and  7,  because  7  is  assumed  to  be  equal  to  90°,  while 
no  such  assumption  is  made  concerning  a. 

Permuting  (1)  in  the  oblique-angled  triangle,  we  have 


cos  a  =  COS  6  cos  c  +  sin  6  sin  c  cos  a, 
cos  h  —  cos  a  cos  c  +  sin  a  sin  c  cos  P, 
cos  c  =  cos  a  cos  6  +  sin  a  sin  b  cos  7. 


(2) 


> 


Eq.  (1)  is  called  the  fundamental  equation  of  spherical  trig- 
onometry, since  all  the  other  formulas  may  be  derived  from  it. 

122.   The  Cosine  of  Any  Angle  of  a  Spherical  Triangle  is  equal 
to  the  product  of  the  sines  of  the  other  two  angles  multiplied  by  the 
cosine  of  their  included  side,  diminished 
by  the  product  of  the  cosines  of  the  qther 
two  angles.  —  We  have 

cos  a  =  cos  b  cos  c  -\-  sin  b  sin  c  cos  a.    (1) 

Since  the  angles  of  the  polar  triangle 
are  the  supplements  of  the  sides  oppo- 
site in  the  original  triangle,  and  vice 
versd,  we  have 

a  =  180°  -a',  b  =  180°  -  yS',  c^  180°  -  7',  a  =  180°  -  a'. 

Substituting  in  (1), 

cos  (180°  -  a')  =  cos  (180°  -  yS')  cos  (180°  -  7') 

+  sin  (180°  -  13')  sin  (180°  -  7')  cos  (180°  -  a'), 

(—  cosy8')(—  cos  7')  4-  sinyS'  sin  7'  (~  cos  a'). 

—  cos  yS'  cos  7'  -1-  sin  /3'  sin  <y'  cos  a'. 


or 


—  cos  a' 


cos  a' 


This  formula  expresses  a  relation  between  the  elements  of  the 
polar  triangle ;  but,  since  the  polar  may  be  any  spherical  tri- 
angle, it  expresses  the  value  of  the  cosine  of  an  angle  of  any 
spherical  triangle.     Dropping  the  primes  and  permuting. 


cos  a  =  —  cos  p  COS  7  +  sin  p  sin  -y  cos  a, 
cos  P  =  -  cos  a  cos  Y  +  sin  a  sin  -y  cos  6, 
cos  -y  =  -  cos  a  cos  p  +  sin  a  sin  p  cos  c,  J 


(2) 


GENERAL   FORMULAS.  188 

123.  The  Sine  Proportion.  —  The  sines  of  the  sides  of  a 
spherical  triangle  are  to  each  other  as  the  sines  of  the  opposite 
angles. 

First  Proof .*  —  From  (1),  Art.  121, 

sin  b  sin  c  cos  a  =  cos  a  —  cos  b  cos  c. 
.  • .  sin^  6  sin^  tf  cos^  a  =  cos^  a  +  cos^  b  cos^  c—2  cos  a  cos  b  cos  c. 
.  • .  sin^  b  sin^  c  ( 1  —  sin^  a)  =  cos^  a  +  cos^  b  cos^  c—2  cos  a  cos  6  cos  c. 
.  • .   sin^  b  sin^  e  sin^  a  —  sin^  6  sin^  e  —  cos^  a  —  cos^  b  cps^  <? 

+  2  cos  a  cos  6  cos  c 
=  (1  —  cos^  ^)  (1  —  cos^  e)  —  cos^a  —  cos^  b  cos^  <?  +  2  cos  a  cos  5  cos  c 
=  1  —  cos'-^  b  —  cos^c  —  cos^  a  4-  2  cos  a  cos  b  cos  <?. 
Dividing  both  sides  by  sin^  a  sin^  b  sin^  c, 

sin^  rt  _  1  —  cos^  a  —  cos^  6  —  cos^  c  +  2  cos  a  cos  b  cos  g 
sin^  a  sin^  a  sin^  6  sin^  <? 

Permuting, 

sin^  /S  _  1  —  cos^  b  —  cos^  a  —  cos^  <?  +  2  cos  ^  cos  a  cos  c? 
sin^  6  sin^  b  sin^  a  sin^  c 

sin^  7  _  1  —  cos^  c  —  cos^  b  —  cos^  a  -\-  2  cos  g  cos  ^  cos  a 
sin^  e  sin^  c  sin^  6  sin^  « 

The  second  members  of  the  three  equations  are  identical. 

sin^«  _  sin^  /3  _  sin^  7 
sin'^a      sin^i       sin^c 

or  sing  _ sin p _ sin y^  ^^n 

sin  a     sin  6     sine*  ^ 

Second  Proof. — From  any  point  A  on  OA  pass  the  planes  AD  C 
and  ABB  perpendicular  to  OC  and 
OB^  respectively,  and  let  AB  be  their 
line  of  intersection.  Then  AB  will  be 
perpendicular  to  the,  plane  BOO^  being 
the  intersection  of  two  planes  perpen- 
dicular to  BOO,  and  BB  and  BO  will 
be  perpendicular  to  OB  and  0(7,  re- 
spectively. The  triangles  ABO  and 
ABB  will  be  right-angled  at  D,  and  F,e  102. 

*  Or  find  cos  a  from  (1),  Art.  121 ;  then  sin^a  =  1  —  cos^a,  etc. 


134 


SPHERICAL   TRIGONOMETRY. 


the  triangles  AGO  and  ABO  will  be  right-angled  at  C  and  B. 

Also  ABB  =  fi  Siud  ACB  =  7.     Then 

AB  =  AB  sin  ABB  =  AB  sinyS, 

and  AB==AO  sin  AOB  =  AC  sin  7. 

.-.   ^ J5  sin yS  =  J. 6^ sin 7 

But  AB  =  OAsinc, 

and  ^C=  (9 J.  sin  i. 

.  • .    OA  sin  c  sin  >Q  =  (9^  sin  b  sin  7. 

sin  yS  _  sin  7 
sin<? 

sin  /9  _  sin  7 
sin  b      sin  c 


Fig.  103. 


Permuting, 


sin  5 
sin« 


sin  a 


O) 


124.    Additional  Formulas.  —  We  have 

cos  b  =  cos  a  cos  c  +  sin  a  sin  <?  cos  /S. 
sin  a  sin  c?  cos  /S  =  cos  6  —  (cos  b  cos  <?  -f  sin  b  sin  c  cos  «)  cos  <? 
=  cos  b  —  cos  6  cos^  c  —  sin  5  sin  c  cos  c  cos  a 
=  cos  b  sin^  c  —  sin  b  sin  <?  cos  c  cos  a. 
.•.  sin  a  cos  p  =  cos  6  sin  c  -  sin  ft  cos  c  cos  a,  (1) 

Applying  (1)  to  the  polar  triangle  and  dropping  the  primes, 
sin  a  cos  6  =  cos  p  sin  7  +  sin  p  cos  7  cos  a,  (2) 

Dividing  (1),  member  for  member,  by  the  equation 
sin  a  sin  yS  =  sin  b  sin  «, 


we  have 


COtyS 


cot  b  sin  c  —  cos  c  cos  a 


sin  a 
.*.  sin  a  cot  p  =  cot  6  sin  c  -  cos  c  cos  a. 

Transposing, 

sin  c  cot  b  —  sin  «  cot  yS  +  cos  c  cos  a. 
Permuting, 

sin  a  cot  6  =  cot  p  sin  v  4-  cos  a  cos  -y. 
Other  formulas  may  be  found  by  permuting  (1),  (2),  and 
(3).     Among  these  are  the  following  : 

from  (1),     sin  a  cos  7  =  cos  c  sin  b  —  sin  c  cos  b  cos  «;  (5) 

from  (3),     sin  a  cot  7  =  cot  c  sin  5  —  cos  5  cos  a,  (6) 

and  sin  7  cot  a  =  cot  a  sin  b  —  cos  6  cos  7.  (7) 


(3) 


(4) 


CHAPTER  X. 

RIGHT  SPHERICAL  TRIANGLES. 

125.  Formulas  for  Right  Spherical  Triangles.  —  The  follow- 
ing equations  have  been  shown  in  Chap.  IX  to  be  true  for 
all  spherical  triangles  : 

cos  c  =  cos  a  cos  h  -f  sin  a  sin  h  cos  7,  (a) 

cos  a  =  —  cos  /3  cos  7  +  sin  y8  sin  7  cos  «,  (])) 

cos  7  =  —  cos  a  cos  ^S  +  sin  a  sin  /S  cos  c,  (<?) 

sin  a      sin  5       sine  ^7^ 

-: —  =  — — -z  —  — — 1  \.^) 

sin  a      sinyS      sin  7 

sin  a "(308  7  =  cose  sin 6  —  sine  cos  6  cos  a,  (^) 

sin  7  cot  a  =  cot  a  sin  b  —  cos  6  cos  7.  (/) 

By  making  7  =  90°  we  get  seven  formulas  applicable  to  right 
triangles,  and  by  permuting  these  three  others  are  found. 


From  (a), 

cos  c  =  cos  a  cos  6. ' 

^ 

(1) 

From  (c), 

cos  c  =  cot  a  CQt  p. 

(2) 

From  (^), 

cos  «  =  sin  yS  cos  a. 
cos  yS  =  sin  «  cos  h.   . 

(3) 

Permuting, 

From  (e), 

cos  a  =  tan  h  cot  c. 

cos  y8  =  tan  a  cot  c.  . 

1 

(4) 

Permuting, 

From  ((^), 

sin  a  =  sin  <?  sin  a. 

i 

(5) 

From  ((?), 

sin  5  =  sin  c  sin  /3.  . 

^''''■' 

From  (/), 

sin  b  =  tan  a  col  a. 
sin  a  =  tan  b  cot  yQ.  . 

(6) 

Permuting, 

135 

136 


SPHERICAL   TRIGONOMETRY. 


126.  Formulas  for  Right  Spherical  Triangles.  Geometrical 
Proof.  —  Let  OB  be  unity.  From  B  pass  the  plane  BAC  per- 
pendicular to  OA,  Then  AB  and  A  Q  are  perpendicular  to  OA, 
and  OB  is  perpendicular  to  00. 

.'.  OB  =  iima,  00=  cos  a,  AB  =  smc, 
OA  =  cos  c,  OAB  =  a. 

.*.   AO  =  OB  cot  a  =  sin  a  cot  a,         (a) 

AO  =  AB  cos  a  =  sin  c  cos  a,  (6) 

AO  =  00 sin  h  =  cos  a  sinh^  ■        (<?) 

^  (7  =  OJ.  tan  5  =  cos  tf  tan  h.         (d) 

^Equating  these  values  of  A  0,  we  obtain  the  following  for- 
mulas : 


From  (a)  and  (^),  sin  a  =  sin  c  sin  a. 

Permuting,  sin  i  =  sin  c  sin  /S. 

From  (a)  and  (t-),  sin  5  =  tan 

Permuting,  sin  a  =  tan 


a  cot  a.  1 

h  cot  yS.  1 


sin  a 


From  (a)  and  (^?),       cos  c  = cot  a  ; 

tan  6 

cos  c  =  cot  a  cot  /3. 
sin  5 


.-.   from  (6), 
From  (i)  and  ((?), 


cos  a  =  cos  a 


sni  d? 


.-.   from  the  sine  proportion  or  from  (5) 

cos  a  =  cos  a  sin  y8.  "[ 
cos  yS  =  cos  b  sin  a.  J 


Permuting, 

From  (^)  and  (rZ), 
Permuting, 

From  ((7)  and  (cZ), 


cos  a  =  tan  5  cot  c. 
cos  /3  =  tan  a  cot  e. 

cos  (?  =  cos  a  cos  5. 


(5) 
(6) 

(2) 


(3) 
0) 


127.  Napier's  Rules.  —  Napier,  the  celebrated  Scotch  mathe- 
matician, devised  two  rules  by  which  the  ten  formulas  connect- 
ing the  elements  of  a  right  spherical  triangle  may  be  easily 
written. 


\^ 


RIGHT   SPHERICAL   TRIANGLES. 


137 


Fig.  106. 


He  called  the  sides  a  and  h  about  the  right  angle,  and  the 
complements  of  the  two  oblique  angles 
and  of  the  hypotenuse,  the  parts  of  the 
triangle,  not  considering  the  right  angle 
as  a  part ;  the  parts,  then,  are  a,  5, 
90°  -  c,  90°  -  «,  90°  -  /9,  which  we 
shall  call  a,  5,  c\  «',  /S'.  By  reference 
to  the  circular  figure,  in  which  the  parts 
are  arranged  in  their  order  in  going 
around  the  triangle,  it  will  be  seen  that 
if  any  three  parts  are  considered,  either 
one  will  lie  between  the  two  others, 
being  adjacent  to  both,  or  one  will  be 
separated  from  the  other  two  by  inter- 
mediate parts.  Thus  h  lies  between  a' 
and  a,  being  adjacent  to  both,  and  /3^  is 
separated  from  both  a'  and  h.  The  part 
which  lies  between  two  others  adjacent 
to  it  or  is  separated  from  both  the  others 
by  intervening  parts,  is  called  the  middle  part;  the  two  others, 
if  adjacent  to  it,  are  called  adjacent  parts^  and  if  separated, 
opposite  parts.  Thus,  if  c\  a'<,  and  h  are  considered,  a'  is  the 
middle  part,  and  c'  and  b  are  the  adjacent  parts  ;  if  c\  ff^  and  b 
are  considered,  b  is  the  middle  part,  and  c'  and  ^'  are  the  oppo- 
site parts. 

Napier's  rules  are  : 

1.  The  sine  of  the  middle  part  is  equal  to  the  product  of  the 
tangents  of  the  adjacent  parts. 

2.  The  sine  of  the  middle  part  is  equal  to  the  product  of  the 
cosines  of  the  opposite  parts. 

The  rules  may  be  easily  remembered  by  the  a  in  the  words 
tangent  and  adjacent  and  the  o  in  cosine  and  opposite. 

If,  in  a  right  triangle,  any  two  elements  besides  the  right 
angle  are  given,  the  other  elements  may  always  be  expressed  in 
terms  of  these  two  by  Napier's  rules.  Thus,  let  the  given  ele- 
ments be  a  and  c. 

(1)  To  find  a  ;  of  the  three  parts  a,  <?',  and  «',  a  is  the 
middle  part,  and  c'  and  a'  are  the  opposite  parts. 


138 


SPHERICAL    TRIGONOMETRY 


.  •.   sin  a  =  cos  c^  cos  a'  =  cos  (90°  —  c)  cos  (90°  —  «)  =  sin  c  sin  «. 

sin  a 


.*.   sina  = 

sine 

(2)  To  find  yS ;    of  the  three  parts  «,  c\  and  /3',  /8'  is  the 
middle  part,  and  a  and  <?'  are  the  adjacent  parts. 

.  ••  sin  yS'  =  tan  a  tan  c'. 

.  • .   sin  (90°  -  yS)  =  tan  a  tan  (90°  -  c) . 

.-.   cosyS  =  tana  cote. 

(3)  To  find  h  ;    of   the  three  parts  a,  c\  and  5,  <?'  is  the 
middle  part,  and  a  and  h  are  the  opposite  parts. 

.-.   sin  e' =  cos  a  cos  6. 

.  •.  sin  (90°  —  c)  =  cos  a  cos  h. 

.'.   cos  c  =  cos  a  cos  6. 

cose 


.-.     COS  5  = 

COS  a 

128.  Species. — Two  angular  quantities  are  said  to  be  of 
the  same  species  when  both  are  less  or  both  greater  than  90°, 
i.e.  when  they  are  in  the  same  quadrant ;  and  of  different 
species  when  they  are  in  different  quadrants. 

129.  Rules  for  Species  in  Right  Spherical  Triangles. 

(1)  An  oblique  angle  and  its  opposite  side  are  always  of  the 
same  species.     From  Napier's  rules, 

sin  h  =  tan  a  cot  a. 

But  sin  h  is  always  positive,  and  therefore  tan  a  and  cot  a  must 
have  the  same  sign  ;  if  they  are  both  positive  a  and  a  will  be  in 
the  first  quadrant,  and  if  both  are  nega- 
tive a  and  a  will  be  in  the  second  quad- 
rant. 

(2)  If  the  hypotenuse  is  less  than 
90°,  the  tivo  oblique  angles  (and  therefore 
the  two  sides')  are  of  the  saine  species; 
if  it  is  greater  than  90°,  the  two  angles 
(and  therefore  the  two  sides)  are  of  dif- 
ferent species.     From  Napier's  rules, 

cos  c  =  cot  a  cot  yS  =  cos  a  cos  b. 


RIGHT  SPHERICAL  TRIANGLES.  139 

If  c  is  less  than  90°  its  cosine  will  be  positive  ;  cot  a  and  cot/8 
must  therefore  have  the  same  sign,  and  hence  a  and  y8  must  be 
in  the  same  quadrant.  If  c  is  greater  than  90°  its  cosine  will 
be  negative  ;  cot  a  and  cot  fi  must  therefore  have  different 
signs,  and  hence  a  and  /9  must  be  in  different  quadrants. 

Thus,  if  a  =  40°  and  ^  =  60°,  a  and  b  must  be,  from  the 
lirst  rule,  in  the  first  quadrant ;  and,  since  a  and  ff  (or  a 
and  h')  are  in  the  same  quadrant,  c  must  be,  from  the  second 
rule,  in  the  first  quadrant.  If  a  =  70°  and  c  =  110°,  from  the 
second  rule  we  see  that  yS  must  be  in  the  second  quadrant,  and 
from  the  first  rule  that  a  is  in  the  first  and  h  in  the  second 
quadrant. 

130.  Solution  of  Right  Spherical  Triangles.  —  There  are  six 
possible  cases,  all  of  which  may  be  solved  by  Napier's  rules  : 

I.  Given  the  hypotenuse  and  an  angle. 

II.  Given  the  hypotenuse  and  a  side. 

III.  Given  the  two  angles. 

IV.  Given  the  two  sides. 

V.    Given  an  angle  and  the  adjacent  side. 
VI.    Given  an  angle  and  the  opposite  side. 

The  required  elements  should  always  be  determined  directly 
from  the  given  elements. 

First  write  the  three  formulas,  each  containing  the  two 
given  elements  and  one  required  element ;  arrange  the  three 
formulas  for  logarithmic  computation  ;  and  then  write  the 
values  of  the  functions  in  their  proper  places,  being  very  care- 
ful about  writing  n  after  the  logarithms  of  the  negative  func- 
tions. If  the  number  of  negative  factors  is  even,  the  result 
will  be  positive  ;  if  it  is  odd,  the  result  will  be  negative  and 
n  should  be  written  after  the  resulting  logarithm. 

If  the  sine  of  a  quantity  is  found  by  the  computation  to 
be  positive,  the  quantity  may  be  either  in  the  first  or  in  the 
second  quadrant,  the  proper  quadrant  being  determined  by  the 
rules  for  species ;  if  the  sine  is  negative  the  triangle  is  impos- 
sible, since  the  elements  of  the  triangle  are  each  less  than  180°. 
If  the  cosine,  tangent,  or  cotangent  is  found  to  be  positive,  the 
quantity  lies  in  the  first  quadrant ;  if  negative,  the  quantity 
lies  in  the  second  quadrant. 


140 


SPHERICAL   TRIGONOMETRY. 


A  check  formula  in  each  case  is  found  by  applying  Napier's 
rules  to  the  three  unknown  elements  ;  thus,  if  a  and  h  are 
given,  a,  /S,  and  c  will  be  computed,  and  the  check  formula  is 

cos  e  =  cot  a  cot  ff, 
131.    Case  I.    Given  the  Hypotenuse  and  an  Angle. 


1.    c?  =  129M4'.6, 


43°  15'. 7. 

To  find  ff  ;   cos  c  =  cot  a  cot  ^  ; 


.-.   cotyS 


cose 


cot  a 
To  find  b  ;    cos  a  =  tan  b  cot  c  ; 


.-.   tan  6 


cos  a 


cot  c 
To  find  a  ;    sin  a  =  sin  a  sin  c. 
Check  :  sin  a  =  tan  b  cot  /S. 


logcosc  =  9.80114/1 
logcot  a  =  0.02637 

logcot/3  =  9.77477  ?t 
.-.  /3  =  120°  46'.03 


log  cos  a  =  9.86227 
logcotc  =  9.91214  n 

log  tan  &  =  9.95013  n 
b  =  138°  16'.96 


log  sin  o  =  9.83590 
+  log  sine  =  9.88900 


log  sin  rt  =  9.72490 
a  =32°  3'.  4 


By  the  rules  for  species  j8  must  be  in  the  second,  a  in  the  first,  and  b  in  the 
second  quadrant. 


\y    2.   c  =  110' 


/3  =  48°28'.6. 

.  a  =  118°46M;  6  =  44°  42'. 


111°7'.2. 


132.  Case  II.    Given  the  Hypotenuse  and  a  Side. 

V    1.    c=    75°    0'.4,  a  =  32°56'.        .-.  b=    72°    2'.8  ;  a  =  34°  15'.0 ;  /3  =  80°    0'.6. 
.r^.    c  =  100°12',      6  =  40°30'.3.    .-.  a  =  103°28M  ;  a  =  08°  50'.5  ;  ^3  =  41°  17'.7. 

133.  Case  III.    Given  the  Two  Angles. 

1.  a=  30°51'.2, /3=  71° 36'.  .-.  a  =  25"12'.8;6=  52°  0'.75;c  =  56°  9'.6. 

2.  0  =  130° 20',  /3  =  100°10'.9.  .-.  a  =  131°  7'.0  ;  6  =  103°24'.5^  c=8in3'.7. 

134.  Case  IV.    Given  the  Two  Sides. 

1.  a=    43°  20',  6  =  74°  13'.    .-.  c  =  78°  35'.3  ;  a  =  44°  26'.0  ;  iS  =  79°  1'.4. 

2.  a  =  100°,    6  =  98°  20.    .-.  c  =  88°  33'.5  ;  a=99°53'.8;  /3  =  98°  12'.5. 

135.  Case  V.    Given  One  Side  and  the  Adjacent  Angle. 


1.  6  =  66°  29',  a  =  50°  17'. 

2.  a  =  24°  41',  /3=  140°  34'.  7. 


a=47°40'.5;  c=  74°  27'.6;  /3  =  72°  7'.5. 


b  =  161°3'.2 


149°15'.0:  a  =  54°45'.6. 


RIGHT   SPHERICAL   TRIANGLES. 


141 


136.    Case  VI.     Given  One  Side  and  the  Angle  Opposite. 

Let  a  and  a  be  the  given  elements. 

To  find  b  ;    sinh  =  tan  a  cot  a. 

To  find  c  ;    sina  =  sin  (?  sina  ;  '  .*.  sine 


sma 


To  find  /3  ;  cos  «  =  cos  asin/S  ;   .  *.   sin  yS  = 


sma 
cos  a 
cos  a 


All  three  quantities  are  found  by  their  sines ;  each  of  the 
three  quantities  may  therefore  be  in  either  the  first  or  the 
second  quadrant,  and  there  will  be  two  solutions.* 

If  P  is  the  pole  of  AC,  PC  will  be  perpendicular  to  AG^ 
and  the  distances  BO  will  be  equal  if  m 
is  equal  to  n.  Either  of  the  triangles 
AB  0  will  satisfy  the  conditions  of  the 
problem,  being  right-angled  at  O  and 
having  A  =  a  and  B0=  a. 

The  rules  for  species  must  be  applied 
in  determining  which  values  belong  to 
each  solution.  Thus,  if  a  is  greater 
than  90°,  and  /Q  in  the  first  solution  is 
less  than  90°,  c  must  be  greater  than 
90°  in  that  triangle  ;  in  the  second 
triangle,  where  y3  is  greater  than  90°,  c  must  be  less  than  90°. 
Each  side,  of  course,  is  of  the  same  species  as  its  opposite 
angle.     These  results  may  be  written  : 

First  solution  :       «  >  90°  ;  /3  <  90°;  <?  >  90°  ;  a  >  90°  ;  h<  90°. 

Second  solution  :  «  >  90°  ;  yS  >  90°  ;  c<90°;  a  >  90°  ;  b>  90°. 

1.    a  =  160^  12'.2,  a  =  150°  37'. 

log  sin  a  =  9.52979 
log  sin  a  =  9.69077 

log  sin  c 


Fig.  109. 


log  tan  a  =  9.55625  n 
+  log  cot  a  =  0.24942  n 

log  sin  b  =  9.80567 
b=   39''44M 
b'  =  140°  15'.9 


log  cos  a  =  9.94020  n 
log  cos  a  =  9.97354/1 


9.83902 
c'=    48°39M 
c  =  136°  20' .9 


log 

sin|3  = 

9.96666 

/3  = 

67°  50' 

2 

/3'  = 

112°   9' 

8 

b  =  40°  50',  /3  =  62°  14'.  .:  a  =  27° 3'.9  ;  a  =  38° 0'.4  ;  c  =  47° 38'.6  ; 

or  a'  =  152°  56'.  1  ;  a'  =  141°  59'.6  ;  c'  =  132°21'.4. 


•  The  triangle  is  supposed  to  be  possible.     The  two  solutions  are  identical 
when  a  =  o. 


142 


SPHERICAL   TRIGONOMETRY. 


137.    Special  Cases. 

1.  C  =  90°,  a  =  90^ 

2.  c  =  90°,  a  =  90^. 

3.  a  =  90°,  /3  =  90°. 

4.  «  =  90°,  &  =  90°. 
6.  a  =  90°,  i3  =  90°. 
6.  a  =  20°,  a  =  20°. 


.  a  =  90°  ;  b  and  /3  indeterminate. 

.  a  =  90°  ;  b  and  /3  indeterminate. 

•.  rt  :=  90°  ;  b  =  90°  ;  c  =  90°. 

.  c  =  90°  ;  a  =  90°  ;  /3  =  90°. 

.  c  =  90°  ;  6  =  90°  ;  a  =  90°. 

■.  c  =  90°  ;  &  =  90°  ;  /3  =  90°. 


138.    Additional  Examples. 

1.  rt  =  40°42'.4,  c  =  63°20'. 

.-.  6=53°41'.9;  a  =  46°  52'.25  ;  /3  =  64°24'.0. 

2.  a  =  70°15'.5,  a  =  81°42'.7. 

.:   b=    23°57'.0;  ^=  25°  15'.7  ;  c  =  72°  1'.25 ; 
or  &'  =  156°  3'.0  ;  ^'  =  154°  44'.3  ;  c'  =  107°  58'.75. 

3.  6  =  30°32'.4,  a  =  36°  44'. 

.-.  a=20°46'.0;  c  =  36°21'.6;  /3  =  58°59'.7. 

4.  c  =  72°  10',  a  =  30°  43'. 

.'.  a  =  29°  5'.0  ;    &  =  69°  29'.0 ;   j8  =  79° 41'.25. 
6.    0  =  106° 34- .2,    /3  =  33°11'.7. 

.-.  a  =  121°23'.6;   6  =  29°  ll'.O  ;   c  =  117°  3'.0. 

6.  a  =  28°  47',    &  =  110°27'.3. 

.-.   c  =  107°50'.2;    a  =  30°  23'.1 ;   /3  =  100°  10'.9. 

7.  c  =  54°12'.2,    )S  =  164°50'.4. 

.-.   a  =  99°0'.3;    6  =  167°45'.2;    a  =  126°45'.9. 

8.  a  =  40°  8',    i3  =  74°30'.2. 

.-.   6  =  66°43'.5;  c  =  72°25'.0;    a  =  42°32'.7. 

W    9.    c  =  102°30',    a  =  125°13'.4. 

.-.  «  =  127°8'.l;    6  =  68°49'.0;   /3=72°49'.8. 

10.  a  =  40°42'.4,    ^  =  67°51'.0. 

.-.   a  =  35°4'.4;    ?>  =  54°42'.0;    c  =  61°46'.6. 

11.  6  =  163°14'.2,    c  =  112°41'.8. 

.  •.  rt  =  m°  14'.1  ;   a  =  82°  45'.75  ;   /3  =  161°  46'.9. 

12.  a  =  120°  30'.2,    b  =  140°  12'. 

.-.  c  =  67°2'.8;    a  =  110°  39'.7  ;   /3  =  135°  57'.7. 

13.  c  =  50°  20'.2,    /3  =  101°  29'.4. 

.-.  rt  =  166°29'.5;   ?>  =  131°  1'.7  ;    a  =  162''20'.1. 

14.  a  =  82°4'.4,    /3  =  8°22'.3. 

.-.  a  =  18°42'.2:   c  =  18°  53'.25  ;    Zy  =  2°  43' or  2°  44'. 


RIGHT  SPHERICAL   TRIANGLES. 


143 


15.  a  =  130°40'.7,  c=  76°  31 '.6. 

.-.  6  =  112°33'.0;  a  =  128°26'.6;  /3  =  107"  28'. 76. 

16.  6  =  10°10'.2,  /3=  16°40'.6. 

.:  a=    39°  43'.9  ;  c  =  40°  48'. 1  ;  o  =  78°  0'.7  ; 
or  a'  =  140°  16M  ;  c'  =  139°  11'.9  ;  a'  =  101°  59'.3. 

17.  6  =  67°8'.3,  a  =  104°16'.2. 

.-.  a  =  106°  50'.8  ;  c  =  99°  2'.8  ;  ^  =  58°  16'.4. 


18.  a  =  20°  64',  b  -  6^°  26'.7. 


.-.  c  =  m°  14'.  1 ;  a  =  22°  66'.5  ;  /3  =  80°  19'.  2. 


139.    Isosceles  Triangles.  —  If  an  arc   of  a  great  circle  be 
drawn  from  the  vertex  perpendicular  to  the  base,  it  will  bisect 
both  the  base  and  the  angle  at  the  ver- 
tex, dividing  the  triangle  into  two  equal 
right  triangles  that  may  be  solved  by 
Napier's  rules. 

1.  a  =  110°47'.3,  y8=92°14'.6. 

.-.  -l-y8  =  46°7^3. 

To  find  a:         cosa=  cot  «  cot  JyS  ; 

,  cos« 

.-.   cot  a  = 


Fig.  110. 


COt-l-yS 

To  find  J h  :  sin \h  =  sin  a  sin ^  ff. 

To  find  jt?:    cos-J/S  =  tanj?  cota;   .-.  tanjo  = -^ — 

Check  :  sin  J5  =  tanjt?  cot  a. 


log  cos  a  =  9.55013  n  log  sin  a  =  9.97076  log  cos  ^  /3  =  9.84081 

log  cot  I  /S  =  9.98299  f  log  sin  ^  /3  =  9.85783  -  log  cot  a  =  9.57936  n 

log  cot  a  =  9.56714  n  log  sin  ^b  =  9.82859  log  tanp  =  0.26145  n 

110°  15'.54  I  b  =  42°  22'.1  p  =  118°  42'.6 

■ — •  b  =  84°  44'.2                              


2.  a  =  82°  26',  /3  =    64°  42'. 

3.  &  =  56°41',  /3  =  112°44'.6. 


.-.  a=    77°53'.6;  \b=    31°32'.75. 

.-.  a=    38°69'.6;      a=    34°45'.6; 
or  a'  =  141°  0'.4  ;      a'  =  146°  14'.4. 


144 


SPHERICAL   TRIGONOMETRY. 


140.  Quadrantal  Triangles.  —  The  polar  of  a  quadrantal  tri- 
angle is  a  right  triangle  whose  angles 
are  the  supplements  of  the  sides,  and 
whose  sides  are  the  supplements  of 
the  angles,  of  the  original  triangle. 
^90°  We  may  therefore  solve  the  polar  by 
Napier's  rules,  and  then  find  the  ele- 
ments of  the  original  triangle  by 
taking  the  supplements  of  the  ele- 
ments of  the  polar. 


A     b' 


Fig.  111. 

1.    ^=90°,   a=  23°14'.7,    b=:  27°14'.6. 
.-.  y  =  90°,  a'  =  156°45'.3,  yS'  =  152°  45'.4 
are  the  elements  of  the  polar  triangle.* 
To  find  c' :      cos  jS'  =  tan  a'  cot  c' ; 


To  find  b' :      sin  a'  =  tan  b'  cot  /3' ; 

To  find  a' :      cos  a'  =  cos  a'  sin  ff. 
Check:  cos  a'  =  cot  c'  tan  b'. 


cot  c' 


tan  b' 


^  cos  13' 
tan  a' 
_  sin  a' 
~cot/3'" 


log  cote' =  0.31594 

log  tan  6'  =  9.30795  n 

log  cos  a'  ■=  9.62389  n 

c'=   64°12'.8 

6'=168°30'.8 

a'  =  114°52'.4 

.-.  7  =  115°47'.2 

.:  p=    11°29'.2 

.:  a=    65°    7'.6 

log  cos  /3'  =  9.94894  n 
log  tan  a'  =  9.63300  n 


log  sin  a'  =  9.59623  log  cos  a'  =  9.96324  n 

log  cot  /3'  =  0.28828  n      +  log  sin  /3'  =  9.66065 


c=    90°,    7 


22'.7, 


I  =  150°  47 
150°26'.2 


90' 
90^ 
90^ 


a  =  121°  30',       /3  =  112°16'.2. 
.:   b  =  108°51'.l:     - 


94°43'.5;     jS  =    99°36'.6. 
123°30'.75;   y=  102°   4'.7. 


a  =  138°47'.8,    6  =  107°54'.9. 

.-.  a  =  142°15'.2;     ^  =  117°50'.25;   7  =  111°  40'.  1. 

a  =  112°   6'. 5,  7=    74°  30'. 

.  •.  6  =._  56°  39'.6  ;      a  =  116°46'.4;     /3  =    53°36'.9. 

6.  c  =    90°,    a  =   83°  20'.6,    /3  =    77°  14'.3. 

.-.  a=    83°30'.3;     &=    77°19'.3;     7=    91°28'.0. 

7.  c=    90°,    a=    94°22'.2,    a  =  108°  13'.3. 

.-.  b=    14°   6'.2;     /3=    13°25'.3;      7=    72°  17'.5; 
or  6'  =  165°  53'.8  ;    ^'  =  166°  34'.7  ;    7'  =  107°  42'.5. 

*  Note  that  a',  /3',  and  c'  are  not  the  parts  of  the  right  triangle,  but  their 
complements. 


RIGHT   SPHERICAL   TRIANGLES. 


145 


141.  Quadrantal  Triangles  may  also  be  solved  by  the  use  of 
Fig.  112,  in  which  B^  one  of  the  vertices  adjacent  to  the 
quadrantal  side,  is  the  pole  of  the  great  circle  MDGN. 

If  the  triangle  has  one  side  less  than  90°,  as  BO  in  the  tri- 
angle ABC,  produce  that  side  to  B. 
In  the  triangle  ACB,  ABC  =90", 
BAC  =  90°  -  «,  ACB  =  180°  -  7, 
AC=b,  (7i>=90°-a,  SindAB  =  l3 
since  AB  =  ABB.  Therefore,  if  any 
two  elements  of  ACB  besides  the 
quadrant  are  given,  we  know  two 
elements  of  the  right  triangle  ACB 
in  addition  to  the  right  angle.  Hence 
we  could  solve  it  by  Napier's  rules, 
thence  obtaining  the  elements  of  ABC. 

If  one  side  of  the  triangle  is  greater  than  90°,  as  in  BUF, 
then  in  the  triangle  GBF  we  have  GF  =  a  -  90°,  GF  =  /3, 
FF  =  6,  FGF  =  90°,  GFF  =a-  90°,  and  GFE  =  7.  If  any 
two  elements  of  BFF  besides  the  quadrantal  side  are  given, 
we  then  know  two  elements  of  the  triangle  GFF  in  addition 
to  the  right  angle.  Hence  we  could  solve  it  by  Napier's  rules, 
thence  finding  the  elements  of  BFF. 


Fig.  112. 


CROCK.   TRIG. 


10 


CHAPTER  XI. 


OBLIQUE   SPHERICAL  TRIANGLES. 


142.    To  find  an  Angle,  having  given  the  Three  Sides. 

(a)     cos  a  =  cos  b  cos  c 

4- sin  5  sin  <?  cos  a; 


,    „^^ cos  a  -  cos  b  cos  c 

•    •    LOs  CI  — ; 9 


sin  6  sin  c 

which  may  be  solved  by  the  use  of  the 
natural  functions. 

(5)*  To    adapt    (1)   to     logarithmic 
computation,  subtract  each  member  from 
unity, 
-i  -j     cos g  — cos 5  cos c_ sin  b  sine  — cos  a -f  cos 6  cosg 

sin  b  sin  c  sin  b  sin  e 

sin  b  sin  c 
Applying  (4)  of  Art.  73, 

cos  u  —  COS  v=  —2  sin  J (u  +  v) sin  J (w  —  v)^ 


we  have    cos  (b  —  c)  —  cos  a  =  —  2  sin  -J  (5  —  e  +  a)  sin  |  (b 


a) 


since  sin  (  —  a;)  =  —  sin  x. 

Let  a-\-b-\-c=^2s', 


+  2  sin  J  (a  +  6  —  f)  sin  J  (a  —  5  +  <?), 


cos  (6 


a  +  6-e  =  2s-2c  =  2(s-e); 
a-6  +  e=2s-25  =  2(s-^.). 

)  —  cos  a  —  2  sin  (s  —  e)  sin  (^s  —  b'). 

c) 


Permuting, 


.  o  1        sin  (s  -  b)  sin  {s 


2l 


sin^^p 


8in«|Y  = 


sin  6  sin  c 

sin  (s  -  a)  sin  (s 

-c) 

sin  a  sin  c 

sin  (s  -  a)  sin  {s  - 

-b) 

sin  a  sin  6 

Compare  with  Art.  99. 
146 


(2) 


OBLIQUE   SPHERICAL  TRIANGLES. 


147 


((?)*  Add  each  member  of  (1)  to  unity. 

cos  a  —  cos  h  cos  c 


1  -h  cos  «  =  1  + 


sin  h  sin  c 


__  cos  a  —  (cos  h  cos  g  —  sin  h  sin  g) 
sin  h  sin  g 

o       o  1         cos  a  —  cos  (h-\-c) 
.'.   2cos2Ja  = : — ; — H^ — ■ — ^• 

sin  0  sin  g 
Applying  (4)  of  Art.  73,  we  have 

cos  a —  cos  (6  H-g)=  —2  sin  J  (a -}- ^  +  <?)  sin  J(a  — 6  — g) 
= +2  sin-Ka  +  J  +  Osin  J(5  +  g  — a). 
Let     a  +  b  +  c  =  2s;     .-.  ^»4-g-a  =  2s-2a=2(«-a). 
.-.  cosa  — cos(6  +  g)  =  2sins  sin(s  — a). 


Permuting, 


^  Sin  6  sin  c 

cos^ip  =  ^'»^^^"<^^-^\ 
^  sin  a  sin  c 

f^a^i    _sing  sin(8-c) 
MIS    nj  —  ;  z     7       • 

*  sm  a  sm  & 


(3) 


(c?)    Dividing  sin^  J  '^  by  cos^  ^  a,  we  have 


Permuting, 


We  may  write 


.     2 1        Sin  (s  -  b)  sin  (s  -  c) 
2  sm  «  sin  (s  -  a) 

2  sm  s  sm  (s  -  o) 

tan«  1  ^  ^  sinCs-CT)sin(g-6)^ 
2  ^  sin  s  sin  («  -  c) 


(4) 


Let 


1  /sin  (s  —  a')  sin  (s  —  6)  sin  (s  —  g") 

tan  i  a  =  -r— 7 ^\/ ^= ^^ ^ -- 

^         sin  (s  —  a)  ^  sin  s 

_    /sin  (8  -  a)  sin  (s  -  6)  sin  (s  —  c) 


Permuting, 


sins 

••*^"«*  =  8in(5-a)- 
**"|P=sin(r-6)' 

*  Compare  with  Art,  99. 


(5) 
(6) 


148 


SPHERICAL   TRIGONOMETRY. 


Note. — The  center  of  the  inscribed  circle  of  a  spherical  triangle  is  the 
point  of  intersection  of  the  arcs  of  great  circles  bisecting  the  angles  of  the 

triangle.  From  this  point  0  draw  the  arcs 
OZ,  OM,  and  ON  perpendicular  to  the 
sides  of  the  triangle. 

.-.  MA  =  AN\  NB  =  BL)  CM=LC. 

.'.  MA  +  NB-h  CM=  s. 

.:  b  +  NB^s,  since  MA  +  CM  =  b. 

M 

Fig.  114.  .  •.   NB  =  s  -  b. 

In  the  right  triangle  OBN,  by  Napier's  rules, 

sin  NB  =  tan  O.Vcot  NBO. 

,-.  tanO-V  =  -^^Hi^=sin(s-6)tan^)8 


cotiV^O 

=  sin  (s  —  b) 


4 


sin  (s  —  a)  sin  (s  —  c) 
sin  s  sin  (s  —  b) 


4 


sin  (s  —  a)  sin  (s  —  6)  sin  (s  —  c) 


sms 


.-.  tan  ON=r. 
Hence  r  is  the  tangent  of  the  radius  of  the  inscribed  circle. 

143.    To  find  a  Side,  having  given  the  Three  Angles, 
(a)     cos  a  =  —  cos  yS  cos  7 

+  sin  /3  sin  7  cos  a. 

.•.cosa  =  ?^^^«5ii5^,  (1) 

sinpsln-y  ^  ^ 

which  may  be  solved  by  the  use  of  the 

natural  functions. 

^        (5)*    To   adapt   (1)    to   logarithmic 

computation,  subtract  each  member  from 

unity. 

-  ^     cos  a  +  cos  8  cos  7 

.-.  1  — cosa  =  l -. — p5— T^- 

sm  p  sin  7 

_     cos  y8  cos  7  —  sin  ff  sin  7  +  cos  a 

sin  yS  sin  7 

o   .   0  1  cos  (/3  +  7)  +  cos  a 

.'.  2sm2Aa= ^^^ — y^. 

^  sm  p  sin  7 

Applying  the  equation  (Art.  73) 

cosw  +  cos  V  =  2  cos  1^(2*  +  t>)cos  J(w  —  v"), 

we  have   cos(y8+7)+cosa=2cos  J  (a  +  /3+7)  cos  J(y8  +  7  — '0 


♦  Compare  with  Art. 


OBLIQUE   SPHERICAL   TRIANGLES. 


149 


Let  a-|-;9  +  7  =  2AS';  .-.  /9  +  7  -  «  =  2aS' -  2a  =  ^(.S'- a). 
.  *.  cos  (/S  +  7)  +  COS  a  =  2  cos  JS  cos  (aS'  —  a). 


Permuting, 


^  sin  p  sin  Y 

gin2  15=  -cosScos(5-p) 
^  sin  a  sin  y 

sin''  - c  =  ~  *®'' '^ ^®*  C-S^  -  7). 


(2) 


sin  a  sin  P 

(c?)*   Add  each  member  of  (1)  to  unity. 

^  ^       cos  a  +  cos  yS  cos  7 

.-.   l  +  cosa  =  lH : — ri—' 

sin  p  sin  7 

_  cos  a  4-  cos  /Q  cos  7  +  sin  ^  sin  7 
""  sin  j3  sin  7 

cos  a  -\-  cos  (yS  —  7) 
sin  /3  sin  7 


2cos^^a  = 


Applying  the  equation  (Art.  73) 

cos  u  +  cos  V  =  2  cos  1^  (t^  4-  v)  cos  J  (m  —  v), 
we  have  cosa  +  cos(^  — 7)  =  2  cos  J(a  +  /Q  — 7)  cos  J(a  — /3  +  7). 
Let  a  +  yS  +  7  =  2AS';  .-.  a  + /3  -  7  =  2aS'- 27  =  2  (aS^  -  7); 

«  -  yg  +  7  =  2aS'- 2/3  =  2(aS'- /3). 
.  •.    cos  a  +  COS  (yS  —  7)  =  2  cos  (^S  —  /3)  cos  (aS'  —  7). 


..  cos  ^a-  sinpsinv 

•r>  ,  .  o  1  ,        COS  (5>  -  a)  cos  (S  -  y) 

Permuting,  cos«  ^  6  =  sinasinY ' 

«i        cos  (S  -  g)  cos  {S  -  P) 
^^^^^=  sinasinp 

(cT)    Dividing  sin^Ja  by  cos^i-a,  we  have 

-cosScos(S  -g) 

cos(«-P)cos(;S-y)' 


(3) 


tan»^a 


Tj  .•  X     oi^        -  cos  aS  cos  (*S  -  p) 

Permuting,  tan«-6  =  c^c^S  -  g)  cos (^  -  y)^ 

tan^i^^     -cos^^cos(^-y)    ^ 

2        cos(-S'-g;cos(S-p)   J 


(4) 


♦  Compare  with  Art.  99. 


150  SPHERICAL  TRIGONOMETRY. 

We  may  write  

tan  i  a  =  cos  {S  -  a}  J  -cosS' 


COS  (S  —  a)  cos  (aS'  —  /3)  cos  (^S—  7)' 


>'C0S(^-a)C0S(>S-P)C0SC^'-Y)*  ^^ 

.•.  tan  I  a  =  JB  cos  (S  -  a). 
Permuting,  tan  1 6  =  U  cos  (5  -  P),  (6) 

tan  I  c  =  iJ  cos  (S  -  7). 

Note.  —Since  the  sum  of  the  angles  2  S  must  be  between  180°  and  540°,  S 
must  be  between  90°  and  270°,  so  that  COSTS'  is  always  negative  and  hence 
—  cos  S  is  always  positive. 

Note.  —  The  center  of  the  circumscribed  circle  of  a  spherical  triangle  is 
the  point  of  intersection  of  the  arcs  of  great  circles  perpendicular  to  the  sides  of 
the  triangle  at  their  middle  points. 

.-.  AN  =  NB;  BL=LC;  CM  =  MA. 

.-.   0AM  =  OCM;  OANz=  OBN;  OCL  =  OBL. 

.'.   0AM  +  OAN-\-  OCL=  S, 

.'.  OCL  =  S-{OAM-\-OAN)  =  S-a. 

In  the  right  triangle  OCL^  by  Napier*s  rules, 

cos  OCL  =  tan  LCcot  OC. 

-.^      tanLC  tania 

.-.  tan  0C  = 7vFrF  = ttt — r 

cos  OCL     cos(/S^— a) 


S-a)\i 


COS /S  cos  (^  —  a) 
cos  (S  —  a)  \cos  («S'— /3)cos  (S  —7) 


4 


-  cosS 


'cos  (6'-a)cos  (S-p)  C9S  (<S-7) 
Fio-  116.  ...  tan  OC=i?. 

Hence  B  is  the  tangent  of  the  radius  of  the  circumscribed  circle. 

144.   Napier's  Analogies. 

(1)    From  (4)  or  (6),  Art.  142, 

tan  ^  «  _  sin  (s  —  b') 
tan  ^ /3  ~  sin  (s  —  a) 

By  division  and  composition, 

tan  1^ «  —  tan  |^  ^  _  sin  (s  —  5)  —  sin  (s  —  a)  ^  ^ 

tan ^ a  +  tan  i  yS  ~  sin  (s  —  6)  +  sin  (s  —  a)' 


OBLIQUE   SPHERICAL   TRIANGLES.  151 

But 

tan  J  a  —  tan  ^  /8  __  sin  | «  cos  ^  /3  —  cos  J  a  sin  |  /S  __  sin  J  (a  —  yS) 
tan  I  «  +  tan  1/5""  sin  ^  «  cos  ^ff-\-  cos  ^  a  sin  ^  /8  ~  sin  |  («  +  /S)' 

Also,  from  (1)  and  (2),  Art.  73, 

sin  (s  —  J)  —  sin  (s  —  a)  __  2  cos  ^  (2  s  —  a  —  6)  sin  ^  (a  —  J) 

sin  (s  —  6)  4-  sin  (s  —  a)      2  sin  ^  (2  s  —  a  —  6)  cos  ^  («  —  i) 

__  tan  ^  (t?  —  6) 

tan  J  c 

Substituting  these  values  in  (a),  and  reversing  the  order, 

tml(a-b)     8iii|(tt-p) 

=  — .  i^i) 

tan^c         siii^(a+p) 

(2)    Substituting  the  values  of  a,  6,  c,  a,  and  yS  in  terms  of 
the  elements  of  the  polar  triangle,  (1)  becomes 

tan  1  (180°  -cc'  -  180°  +  /SQ  ^  sin  |-  (180°  -  a'  -  180°  +  b') 
tan  J  (180°  -  y)  sin  J  (180°  -  a'  +  180°  -  ^'y 

tan  1  (/g^  -  «0  ^  sin  j-  (6^  -  g^) 
cot^y         ~sin^{a'  +  b'y 

-  t^n  1  («^  -  /gp  ^  -  sin  j  (g^  -  6^) 
cot^y  sini(a'  +  ^0   ' 

Changing  the  signs  and  dropping  the  primes, 
taii|(a-p)     sin|(a-6) 


cot|Y  sin|(a  +  6) 

(3)  From  (4),  Art.  142, 

,       -     ,       -  ^      sin  (s  —  <?) 

tanAataniy3  = ^^ ^  .     . 

-^  ^  sm  s 

1  +  tan  |-  a  tan  J  /3  _  sin  s  +  sin.  (s  —  <?) 

*   1  —  tan  J  a  tan  J  /^ "~  sin  s  —  sin  (s  —  <?) 

■r»   .       1  +  tan  I  a  tan  J  /3  _  cos  J  a  cos  -^^  /S  +  sin  J  a  sin  ^  yS 
1  —  tan  |-  a  tan  ^  y3      cos  ^  a  cos  |  y8  —  sin  J  a  sin  J  y8 
_cos^(«  — yg) 
'"cos^(a  +  13} 

Also,  from  (1)  and  (2),  Art.  73, 

sin  8  4-  sin  (s  —  e)  _  2  sin  ^-  (2  s  —  c)  cos  |-  <?  __  tan  J  («5  4-  h} 
sin  «  —  sin  (s  —  c)  ~  2  cos  J  (2  s  —  <?)  sin  |^  (? ""       tan  ^  f? 


(2) 


(*) 


(3) 


152  SPHERICAL   TRIGONOMETRY. 

Substituting  these  values  in  (J),  and  reversing  the  order,. 

tan  l(a  +  b)     cos  |  (a  -  p) 

tan|c  cos^(a  +  P) 

(4)  Passing  to  the  polar  triangle,  (3)  becomes 

tan  1  (180°  -a'  -^  180°  -  ySQ  ^  cos  -|-  (180°  -  a'  -  180°  +  b') 
tan  1  (180°  -  y)  cos  |  (180°  -  a'  +  180°  -  6') 

tan  [180°  -  j-  (tt^  +  /gp]  ^  cos  |-  (5^  -  gQ 

tan  (1)0°  -  I  y)  ~  cos  [180°"^-  i  (a'  +  6')]' 

-  tan  1  (a'  +  ffQ  _     cos  |  (a^  -  ^>0 
cot  ^  y'  ~  —  cos  ^  (a'  +  ^')' 

Changing  the  signs  and  dropping  the  primes, 
tan  I  (tt  +  p)     cos  l(a-b) 

— ^-1 —  =  —i (4) 

cot  ^7         COS  ^  (a +  6) 
Eqs.  (1),  (2),  (3),  and  (4)  are  called  Napier's  Analogies. 

145.    Gauss's  Equations.  —  From  (2)  and  (3),  Art.  142,  we 

have  

.    -  -   -,     sin  (s  —  6)^  /sin  s  sin  (s  —  (?)      sin  (s  —  h')        , 

sin  i  a  cos  i  )8  = ^^ ^\ ■. ^.    ,  ^  = ^ ^  cos  i  7  ; 

^  ^  sine       ^       sin  asm  6  sm^  ^ 

-       .10     sin  Cs  —  a')^  /sin  s  sin  (s  —  <?)      sin(s  — a)        . 

cos  i  a  sin  1  /8= ^ ^\ : ^ — j-^= ^ ^cos  i  7  ; 

^    •       ^  sin  <?       ^      sm  a  sin  0  sin  c  ^ 

,  ,  ^     sin  s^  /sin  (s  —  a)  sin  (s  —  6)      sin  s   .     , 

cos  i  a  cos  A  /S=  ^ — \ ^ ■         ■    \ -=- — sin  i-  7  ; 

^  ^         s\nc  ^  sin  asm  6  sine       ^ 

^_  sin  (s  —  c)    /sin  (s  —  a)  sin  (s  —  5) 

~"       sin  (?       ^  sin  a  sin  b 

sin  (s  —  c)   .    , 

= ^^ sm  1 7. 

sin  (?  "^ 

(1)  sinK«+  /g)  =  ^^nr^'^'''''^^  ~  ^)+  sin  {s  -  a)] 

_  cos  ^  7  sin  [g  —  j-  (g  +  ^)]  cos  |(a  —  6) 

sin  ^  c  cos  ^  tf 
_  cos  J  7  cos  1^  (a  —  J) 

~~  QOS^C    . 

••.  cos|csin|(a  +  p)  =cos|7COs|(a-6).  (1) 


OBLIQUE  SPHERICAL   TRIANGLES.  163 

(2)  COS  K«  +  /3)  =  ^^  [sin  >  -  sin  (»-<;)] 

_  sin  1 7  cos  (g  —  ^  g)  sin  j-  c 
""  sin  ^  (?  cos  J  g 

__  sin  J  7  cos  ^  («  +  ft) 
~~  cos  ^  g 

.•.  cos  I  c  COS  ^  (a  +  p)  =  sin  |-y  cos  ^  (a  +  6).  (2) 

(3)  sin  K«  -  ^)  =  ^^  [sin  (s  -  b)  -  sin  («  -  a)] 

_  cos  1 7  cos  [8  —  j^  (g  +  ^)]  sin  j  (a  —  h') 
"~  sin  1^  c  cos  ^  g 

_  cos  ^  7  sin  i(ct  —  h) 
~"  sin  J  (? 

.•.  sm^csiii|(a-p)  =  cos|7siii|(a-6).  (3) 

(4)  cos^  (a  —  yS)  =  ^   .  ^  ^  [sin  s  +  sin  (s  —  <?)] 

_  sin  J  7  sin  (s  —  |^  (?)  cos  ^  c 
sin  ^  c  cos  I  g 

_sin^7sin  j  (a  +  ^) 
"~  sin^^c 

.«.  sin|ccos|(a-p)  =sm^7siii|(a  +  6).  (4) 

Eqs.  (1),  (2),  (3),  and  (4)  are  known  as  Gauss's  Equations^ 
or  Delambre's  Analogies. 

146.    Rules  for  Species  in  Oblique  Spherical  Triangles. 

(1)  If  a  side  (^or  angle')  differs  more  than  another  side  (^or 
angle)  from  90°,  it  is  of  the  same  species  as  its  opposite  angle 
(or  side). 

We  wish  to  show  that  cos  a  and  cos  a  will  have  the  same 
sign  when  the  difference  between  a  and  90°  is  numerically 
greater  than  the  difference  between  b  and  90°.     In  the  formula 

cos  a  —  cos  b  cos  c  ^i  v 

cos  a  = : — ; — : (1) 

sm  6  sni  c 

the  denominator  is  always  positive,  so  that  the  sign  of  the  frac- 
tion, and  hence  that  of  cos  a,  is  the  same  as  that  of  the  numera- 


154  SPHERICAL   TRIGONOMETRY. 

tor.  But  if  a  differs  more  than  h  from  90°,  cos  a  is  numerically 
greater  than  cos  5,  and  hence  greater  than  cos  h  cos  c,  since  cos  c 
cannot  exceed  unity.  Therefore  the  numerator  has  the  same 
sign  as  cos  a;  i.e.  cos  a  and  cos  a  have  the  same  sign,  so  that  a 
and  a  are  in  the  same  quadrant. 

By  a  similar  process,  using  the  formula 

cos  a  -f  cos  B  cos  7  ,«. 

cos  a  = 7^      .^ 1,  (2) 

sinpsin7 

we  can  show  that,  when  a  differs  more  than  yS  or  7  from  90°, 
a  and  a  are  of  the  same  species. 

Since  two  sides  will  in  general  differ  more  than  the  third 
from  90°,  two  angles  will  in  general  be  of  the  same  species  as 
their  opposite  sides.  Thus,  if  a  =  140°,  h  =  50°,  and  c  =  110°, 
we  see  that  a  and  b  differ  more  from  90°  than  c  does;  therefore 
a  will  lie  in  the  second,  and  /3  in  the  first  quadrant,  while  the 
quadrant  of  7  is  not  determined  by  this  rule. 

2.  Half  the  sum  of  two  sides  must  be  of  the  same  species  as 
half  the  sum  of  the  two  opposite  angles.  —  From  (3),  Art.  144, 

tan  i(a  +  ft)  =  tan  J  c  ^^"f^'^"^^-  (3) 

^  ^  ^  ^     cos  J  («  +  p) 

But  c  must  be  less  than  180°;  hence  ^c  must  be  less  than  90°, 
so  that  tan  ^c  is  positive.  Also,  a  —  /3  must  be  numerically 
less  than  180°;  hence  J(«  —  /S)  must  be  numerically  less  than 
90°,  so  that  cos|(a  — /8)  is  always  positive.  Hence  tan|(a  +  6) 
and  cos^(«H-/3)  must  have  the  same  sign.  But  a -{- b  and 
«-i-/S  must  each  be  less  than  360°;  hence  ^(«  +  ft)  and  ^(«  +  yS) 
must  each  be  less  than  180°,  so  that  they  must  be  in  the  same 
quadrant  in  order  that  tan  ^(a  +  5)  and  cos  J(a  +  yS)  may  have 
the  same  sign.  Thus,  if  |(a  +  yS)  is  in  the  first  quadrant,  its 
cosine  will  be  positive;  the  second  member  of  (2)  will  be  posi- 
tive, and  therefore  J(a  +  5)  must  be  in  the  first  quadrant. 
If  cos  J(a  + /3)  is  negative,  tanj(a  +  ft)  will  be  negative; 
therefore  J(«  +  yS)  and  ^(«  +  ft)  must  both  be  in  the  second 
quadrant. 

In  the  example  under  the  first  rule,  after  a  and  yS  have  been 
computed,  the  quadrant  in  which  7  will  lie  may  be  determined 
by  the  second  rule. 


OBLIQUE  SPHERICAL  TRIANGLES. 


156 


147.   Solution  of  Oblique  Spherical  Triangles.  —  Any  spherical 
triangle  may  be  solved  by  the  use  of  the  following  formulas: 


sin  a  _  sinb  _  sinc^ 
sin  a     sin  p     sin  y 


**''2''=Sill(S-«)' 


tan|a  =  Bcos(S-a)5  R=yj—^^-—^ 


/sin(s- 

-a)sin(s  -ft)  sin(«- 

-c) 

r-^^ 

sins 

„_J 

—  cohS 

eosiS-^)co8iS-y) 


a  ^          ■' 

2  ^           *^^ 

tan|c 

~8in|(a  +  P)y 

tan^(a  +  6) 

_COs|(a-P)   ] 

ten|c 

~COs|(a  +  P)' 

ten|(a-P) 

sin|(a-6)- 

C0t|7 

~sin|(a  +  6) 

tan|(a  +  p) 

cos|(a-5) 

cotl-/ 


COS -(a +  6) 


0) 


(2) 
(3) 

(4) 
(5) 

(6) 
(7) 


iM<' 


There  are  six  possible  cases: 

I.  Given  the  three  sides. 

II.  Given  the  three  angles. 

III.  Given  two  sides  and  the  included  angle. 

IV.  Given  two  angles  and  the  included  side. 

V.  Given  two  sides  and  the  angle  opposite  one  of  them. 

VI.  Given  two  angles  and  the  side  opposite  one  of  them. 


148.  Case  I.  Given  the  Three  Sides 
(a,  bf  c) .  —  Find  the  angles  by  the  for- 
mulas 


_    /sin  (s  —  a)  sin  (s  —  5)  sin  (s  —  c') 
tan  J  a  = 


sin« 
r 


sin  (s  —  a) 
Check  by  the  sine  proportion. 


156 


SPHERICAL   TRIGONOMETRY. 


1.   Solve  the  triangle  when  a  =  114°  43'.3,  6  =  136°  19'. 6,  c  =  43°  18'.5. 

8  =  147°  10'. 7                      col  sin  s  =    0.26598  .-.  log  tan  ha  =  9.89910 

5  -  a  =    32°  27'.4  log  sin  {s-  a)=    9.72970  |  o  =  38°  24'.2 

8-h=    10°51'.l  log  sin  (s- 6)=    9.27478  log  tan  ^ /3  =  0.35402 

s-c  =  103°  52'.2  log  sin  (s  -  c)  =    9.98714  ^  /3  =  66°  7'.6 

2s  =  294°21'.4                          logr2  =  29.25760 -30  log  tan ^ 7  =  9.64166 

a  check.                          logr=    9.62880  i7-23°39'.7 


In  finding  log  tan  ^  a  write  log  r  on  the  margin  of  a  slip  of  paper,  place  it 
above  log  sin  (s  —  a),  and  write  the  difference  opposite  log  tan  ^  o  ;  then  find 
log  tan  ^  j8  and  log  tan  |  7  in  the  same  manner. 

2.   a  =  76°40'.4,  6  =  54°21'.3,   c  =  36°8'.7. 

.  •.  I  o  =  60°  1'.8  ;     ^  ^  =  23°  8'.6  ;    i  7  =  15°  49'.3. 
8.   a  =  124°  34'.9,   b  =  6G°  7'.2,   c  =  109°  43'.5. 

.-.  Aa  =  60°l'.3;     |/3  =  37°0'.8;    ^7  =  49°6'.8. 
4.    a  =  30°17'.6,   6  =  22°14'.4,   c  =  18°51'.8. 

.-.  |o  =  47°55'.0;  ^i3  =  24°8'.5;   ^ 7  =  19° 48'.45. 
6.    a  =  130°46'.0,    b  =  113°21'.4,   c  =  102°  16'.2. 

•.  ^a  =  72°38'.0;   ii3  =  68°9'.6;   ^7  =  66°20'.5. 

149.  Case  II.  Given  the  Three  Angles  (a,  p,  -y).  — Find  the 
sides  by  the  formulas 


B 


-4 


—  GOSS 


COS  (S  -  «)  COS  iS  -  yS)  cos  CS-ryy 
tan  J  a  =  J?  cos  (^S  —  a). 
Check  by  the  sine  proportion. 


b 
Fig.  118. 

1.    Solve  the  triangle  when  a  =  116°  19'.4,  /3  = 

:  83°  19'.  2,  7  =  106°10'.6. 

/S'=152°54'.6 

col  (-COSTS')  =  0.05047 

.-.  log  tan  ^  a  =  0.23789 

S-a=    36°35'.2 

logcos(5'-a)=9.90469 

^a  =  59°  57'.7 

S-p=    69°35'.4 

log  cos  (>S'-^)=  9.54249 

log  tan  i&  =9.87569 

S-y=    46°44'.0 

log  COS  C^*- 7)  =9-83594 

J  &  =  36°  54'.6 

3  .5=  305°  49'.  2 
a  check. 

col  i?2  =  9.33359 
logi22  =  0.66641 
log  i2  =  0.33320 

log  tan  ^  c  =  0. 16914 
Jc  =55°63'.l 

OBLIQUE  SPHERICAL  TRIANGLES.  157 

In  finding  log  tan  J  a,  write  log  B  on  the  margin  of  a  slip  of  paper,  place 
it  above  log  cos  (/i?  — a),  and  write  the  sum  opposite  log  tan  J  a;  then  find 
log  tan  J  b  and  log  tan  J  c  in  a  similar  manner. 

2.  a  =  110°  36'. 4,   /3  =  122°  8'. 7,     y  =  140°  20'.3. 

.-.  Ja  =  4r66'.3;  J6  =  57°57'.5;  Jc  =  68°39'.4. 

3.  a  =  120°60'.6,    /3  =    78°  6'.  1,      7=    81°  12'.3. 

.-.  Ja  =  59°65'.2;  i6=40°40M;  Jc  =  43°23'.4. 

4.  0=    80°  20'.  2,   /3=    73°  46'.  7,   7=    54°  8'.  5. 

.-.  Ja  =  32°23'.6;  J6  =  30°63'.7;  Jc  =  24°l'.7. 
6.    a  =  100°61'.3,   /3=    80°47'.6,   7=    74°3'.3. 

.-.  ia  =  49°22'.4;  J&=41°42'.5;  ic  =  37°41'.6. 

150.  Case  III.  Given  Two  Sides  and  the  Included  Angle 
(6,  c,  a).  —  By  permuting  (6)  and  (7),  Art.  147,  we  have 

tanK^-7)=cotJ„?i^i|^,  (1) 

tanK^  +  7)=cotJ-«5^jl|^.  (2) 

Then  /3  =  K^  +  7)+K/8-7). 

Note  that  the  larger  angle  must  be  opposite  the  larger  side. 
To  obtain  a,  we  permute  (4)  and  (5),  Art.  147  : 

tan^a^tanK^-0;i:fg!g.  (3) 

tan.a  =  tanK*  +  0^^|ig±g.  (4) 

The  agreement  of  the  values  oi  ^a  found  from  (3)  and  (4)  is 
a  check  upon  the  computation.  The  sine  proportion  may  also 
be  used  as  a  check. 

Note.  —  In  using  these  formulas,  the  larger  side  and  the  larger  angle  should 

be  written  first  in  the  expressions  b  —  c  and  /3  —  7.     Thus  for  c  >  6,  (1)  would 

be  written 

*      1  ^        ON         *  1     sin  i  (c  -  6) 
tan  i  (7  -  /3)  =  cot  i  a  -^-i — — ix* 
^^'      ^^  ^    sm  ^  (c  +  &) 

Eq.  (1)  may  be  read :  "  The  tangent  of  half  the  difference  of  the  required 
angles  is  equal  to  the  cotangent  of  half  the  given  angle,  multiplied  by  the  sine  of 
half  the  difference  of  the  given  sides,  ancf  divided  by  the  sine  of  half  their  sum." 


158 


SPPIERICAL   TRIGONOMETRY. 


1.    Solve  the  triangle  when  h  =  105"  14'.8,   c  =  43°  17'.2,   a  =  112° 

6=105°  14'. 8  logcot^a  =  9.82251  log  cot  J  a  = 

c=  43°17'.2        log  sin  i(6-c)  =9.71159 
col  sin  J  (6  +  c)  =0.01658 


J(6  +  c)=  74°1C'.0 

i(6-c)=  30°58'.8 

Ja=  56°23'.7 


log  tan  i  (i3-7)  =9.55068 
K^-7)=19°33'.8 


logtan|(&-c)  =9.77843 
log  sin  I  (/3  +  7)  =9.95566* 
col  sin  ^(P-y)  =0.47514* 

log  tan  I  a =0.20923 
|a=58°17'.8 


log  tan  1(6+ c)=  0.55019 
log  cos  ^  (^  +  7)  =9.63322* 
col  cos  I  (/3-7)  =0.02582* 

logtan|a=0.20923 
|a=58°17'.8 


log  cos  J^  (6  — C): 
col  cos  ^  (6  +  C): 

logtani(/3+7): 

H/3  +  7): 
K/3-7): 


47'.4. 

:  9. 82251 
9.93316 
:0.56677 


:  0.32244 

-64° 

32' 

.9 

19° 

33' 

.8 

84° 

6' 

.7 

:44° 

69' 

.1 

2.  a  =  103°  44'.  7,  6  =  64°  12'.3,  y  =  98°  33'.  8. 

.-.  K«  + /S)  =  82°  37'.0  ;  K«  - /S)  =  16°  19'.0  ;  a  =  98°  56'.0  ; 
/3  =  66°18'.0;  ^c  =  51°45'.3. 

S.  a  =  156°12'.2,  6  =  112°48'.6,  7  =  76°32'.4. 

.-.  K«  +  ^)=120°45'.6;  K«  - /3)=  33°  18'.5  ;  a  =  154°4M; 
)8  =  87°27'.l;  Ac  =  31°  54'. 4. 


151.   Case  III.     Second  Method.     Given  6,  c,  a,  to  find  One 
Element  only. 

(1)   To  find  a  only. 


Let 


cos  a  =  cos  b  cos  c  +  sin  ^  sin  c  cos  a. 

w  sin  M=  sin  c  cos  a, ' 

m  cos  M=  cose. 
cos  a  =  m  (cos  5  cos  ilff -f  sin  5  sin  JkT), 
cos  a  —  m  cos  (5  —  i^f ). 


(1) 


(2) 


(2)   To  find  one  angle  only,  /8  or  7.  —  From  (6),  Art.  124, 
sin  a  cot  7  =  cot  ^  sin  6  —  cos  b  cos  a. 
cot  c  sin  b  —  cos  ^  cos  a      cos  <?  sin  ^  —  cos  b  sin  <?  cos  a 


.'.   cot  7; 


sma 


sm  a  sin  c 


♦  The  functions  of  \{^  -  7)  and  of  K/3+7)  should  be  found  by  using  the 
fraction  from  which  the  decimal  of  a  minute  is  found.     Thus, 


log  sin  1  (^  +  7)  =  9.95561  +  f  |  x  6  =  9.95561  +  5  =  9.95666. 


OBLIQUE   SPHERICAL   TRIANGLES. 


159 


Let 


?n  siniHf  =  sine 
m  cos  M  =  cos  c 


cot  7 


cos  a,  1 
_  m  sin  (b  —  M) 


sin  a  sin  c 


(3) 
(4) 


The  formula  for  cot  yS  may  be  found  by  permuting  h  and  c 
in  (3)  and  (4). 

1.   6  =  105°14'.8, 
To  find  a. 
log  sine =9. 83611 
log  cos  a= 9. 588 11  n 
log  (to  sin  i)/) 

=9.42422  n 
log  sin  Jlf = 9.53499  n 
log  cos  ilf= 9. 97286 
log  cose =9. 86209 
log  tan  i¥  =9.56213  71 
ilf=-20°2'.7 
6  =  105°  14'.8 
ft -itf  =  125°  17^5 
logcos(&  — ilf) 

=9.76173  n 
log  TO =9.88923 
log  cos  a =9. 65096  w 
a=116°35'.7 


c  =  43°17'.2,  o  =  112°47'.4. 
To  find  7. 

(1)  log  sin  c  = 
(3)      log  cos  o= 

log(TOsinJJf) 
(4) 

(7)  log  sin  ilf= 

(8)  log  cos  ilf  = 

(2)  log  cos  c= 

(5)  log  tan  ilf  =9.56213  n 


(6) 
(10) 

(11) 

(12) 

(9) 
(13) 
(14) 


(1) 
(3) 

(5) 
(8) 
(9) 
(2) 
(6) 

Jf  =  -20°2'.7      (7) 
6  =  105°  14'.8    (11) 
6-3/  =  125°17^5    (12) 
log  sin  (  6  — ilf) 


=9.91180 

log  TO =9. 88923 

col  sin  a =0.03530 

col  sine =0.1 6389 

log  cot  7 =0.00022 

7=44°  59M 


(13) 
(10) 

(4) 
(14) 
(15) 
(16) 


To  find  /3. 
log  sin  6  =  9.98444 
log  cos  g = 9.58811  n 
log(TO  sin  M) 

=9.57265  n 
log  sin  iW =9.91 266  n 
log  cos  Jf= 9.76002  n 
log  cos  6  =  9.41992  n 
log  tan  i¥  =0.15263 
JW=234°52'.0 
e=  43°17'.2 
c-ilf=- 191°  34\8 
log  sin  (c—ilf) 

=9.30263 
log  TO  =  9.65989 
col  sin  0=0.03530 
col  sin  6=0.01556 
log  cot  i3= 9.01338 
i8=84°6V7 


2.  a  =  103°44'.7,    6  =  64°  12'.3,   7  =  98°33'.8. 

.-.  ilf  =  211°  19'.8,  c  =  103°30'.6,  a  =  98°56'.0;  Jf=-17°7'.4,  /3  =  66°18'.0. 

3.  a  =  156°12'.2,  6  =  112°  48' .6,  7  =  76°32'.4. 

.-.  iW=174°8'.4,  c  =  63°48'.9,  a  =  154°4'.2;  ilf  =  151°2'.3,  /3  =  87°27M. 


152.   Case   IV.     Given   Two  Angles  and  the  Included  Side 
(a,   p,  c).  —  From  (4)  and  (5),  Art.  147,  we  have 


Then 


4.      1  ^        7N      ^      1     Sin  ^(«  —  /3) 
tani (a  -  ^i)  =  tan ^ c    .    f  ) — -^, 


tan  1  (a  +  ^)  =  tan  J «? 


cosj-(«-y9) 


cos  J  (a  +  yS) 
«  =  K^  +  ^)  +  K«  -  ^)' 


(1) 
(2) 


160 


SPHERICAL   TRIGONOMETRY. 


To  obtain  7,  use  (6)  and  (7),  Art.  147  : 


cot  J  7  =  tan  -|-  (<^  +  /3) 


cos  -|-  (a  +  h') 


(3) 


COS  1^  («  —  6) 

The  agreement  of  the  values  of  ^y 
*^  found  from  (3)  and  (4)  is  a  check  upon 
the  computation.     The  sine  proportion 
may  also  be  used  as  a  check. 
See  the  note,  Art.  150. 
1.    Solve  the  triangle  when  o  =  104°  30'.7,  /3  =  62°  52M,  c  =  66°  6'.4. 


a  =  104°  30'.  7 
i8=   62°52M 


Ka  +  ^)^ 


}°41'.4 


logtan^c  =  9.72665 
logsini(a-  /3)  =  9.55079 
col  sin  i(a  +  ^)  =  0.00264 


^(a  -  ^)  =    20°  49'. 3       log  tan  |(a  -  &)  =  9.28008 


^c=    28°   3'.2 

logtani(a-i3)  =  9.58012 
logsinK«+  &)=  9.98908 
col  sin  |(a- 6)  =  0.72768 

log  cot^7  =  0.29748 
^7=20°45'.2 


^(a-6)  =  10°47'.4 


log  tan  I  (a  +  ^)  =  0.95633 
log  cos  -|(a  +  &)  =  9.33339 
col  cos  Ka -&)  =  0. 00775 

logcot  17  =  0.29747 
|7=26°45'.2 


logtanic  =  9.72665 
logcosi(o-/3)=  9.97066 
col  cos  ^(a  +  py=  0.95897 

log  tan  l(a  +b)=  0.65628 
i(a  +  6)=77°33'.4 
K«-&)  =  10°47'.4 

a=88°20'.8 
6=66°46'.0 


60°43'.6. 

1 


2.  a  =  140°43'.2,  /3  =  100°4'.6,  c 

.-.  |(a  +  6)=132°38'.88 

6  =  119°  22'. 56  ;  ^7  =  40°  7'. 42. 

3.  a  =  140°  24'.6,  ^  =  12°  18'.6,  c  =  28°  7'.7. 

...  ^(rt  ^i))=  24°  55'.9  ;    1  (a  -  &)  =  13° 3'.0  ;   a  =  37°  58'.9  ; 
6  =  11°52'.9;    |7  =  14°36'.7. 

153.    Case  IV.    Second  Method.     Given  p,  -y,  a,  to  find  One 
Element  only. 

(1)    To  find  a  only. 

cos  a  =  —  cos  y3  cos  7  +  sin  /3  sin  7  cos  a. 
Let  m  sin  M  =  sin  7  cos  <x,  1 

m  cos  M  =  cos  7.  J 

.-.  cosa  =  —  m(cos/3cosil!f— sinySsiniJf). 
.-.   cos  a  =  —  w  cos  (3/ + /3) . 


(1) 

(2) 


OBLIQUE   SPHERICAL  TRIANGLES. 


161 


(2)    To  find  one  side  only^  b  or  c.  —  Permuting  (G),  Art. 


124, 


sin  yS  cot  7  =  cot  c  sin  a  —  cos  a  cos  )8. 


,     _  sin  ff  cot  7  -h  cos  a  cos  ^  _  sin  /g  CO87  +  cos  a  cos  )g  sin  7 

sin  a  sin  7 


Let 


sin  a 

wsiniHf  = 
wcosiHf 


=  sin  7  cos  a,     1 
=  cos  7.  J 


•.  cotc  = 


(3) 
(4) 


^msin(il[f+/3) 
sin  a  sin  7 

The  formula  for  cot  b  may  be  found  by  permuting  b  and  c  in 
(3)  and  (4). 


1.     /3  : 

=  140°43'.2, 

•  7  =  ■ 

100°4'.6,  a  = 

=  60°  43'.  6. 

To  find  a. 

To  find  c. 

To  find  h. 

log  sin  7  z 

=9.99325 

(1) 

log  sin  7  : 

= 

(1) 

log  sin /3= 

=  9.80148 

log  cos  a: 

=  9.68929 

(3) 

log  cos  a- 
log  (m  sin 

= 

(3) 

log  cos  a = 
log  (wi  sir 

=9.68929 

log  {m  sin 

M) 

M) 

\M) 

: 

=9.68254 

(4) 

= 

= 

(5) 

= 

=9.49077 

log  sin  ilf= 

=9.97306 

(7) 

log  sin  ilf= 

= 

(8) 

logsin  ilf = 

=9.56977 

log  cos  M  z 

=9.53348  71 

(8) 

logcos  ilf  = 

= 

(9) 

logcositf= 

=  9.96778  n 

log  cos  7  - 

=9.24296  n 

(2) 
(5) 

log  cos  7  : 

logtanilfi 

= 

(2) 
(6) 

logcosi3= 
logtanilf= 

=9.88877  n 

log  tan  i»f  I 

=0.43958  n 

=0.43958  n 

=9.60200  n 

M-. 

=  109°58'.4 

(6) 

M-- 

=  109°58'.4 

(7) 

Jf= 

=  158°  12'.  1 

/3: 

=  140°  43'.  2 

(10) 

^-- 

=  140°  43'.  2 

(11) 

7  = 

=  100°   4'.6 

Jf  +  ^: 

=250°41'.6 

(11) 

M-\-^-- 

=250°  41 '.6 

(12) 

J/+7  = 

=258°  16'.  7 

logcos(ilf+/3) 

logsin(3f+^) 

log  sin  (3/ +7) 

: 

=  9.51933  w 

(12) 

: 

=9.97486  n 

(13) 

= 

=  9.99085  n 

lOg(-Wl): 

=9.70948  w 

(9) 

logm  = 

=9.70948 

(10) 

logm  = 

=9.92099 

log  cos  a  : 

=9.22881 

(13) 

ri4) 

col  sin  a: 

=0.05934 

(4) 

col  sin  a  = 

=0.05934 

a- 

=  80°  15'.0 

col  sin  7 1 

=0.00675 

(14) 

col  sin /S= 

=0.19852 

log  cot  C: 

=9.75043  n 

(15) 

log  cot  6  = 

=0.16970  w 

C- 

=  119°22'.5 

(16) 

h-- 

=  145°55'.2 

2.  a  =  104°  30'.7,  /3  =  62°  52'. 1,  c  =  56°  6'.4. 

.-.   ilf=  114°53'.9,  7=53°30'.5,  a=:88°20'.8;   JW  =  47°  25'.2,  6  =  66°46'.l. 

3.  a  =  140°24'.6,  j8  =  12°  18'.6,  c  =  28°  7'.7. 

.-.  Jf=  143°53'.7,  7i=29°13'.3,  a  =  37°58'.8;  iW  =  10°53'.6,  5  =  11°52'.9. 

4.  a  =  109°  23'. 5,  /3  =  76°  47'.4,  c  =  121°  32'. 8. 

.-.  -Jf=236°4'.l,  7  =  113°51'.9,  a  =  118°28'.5;  i»/=  294°  9'.8,  &  =  65°7'.5. 

CROCK.    TRIG.  —  11 


162 


SPHERICAL  TRIGONOMETRY. 


154. 


Given  Two  Sides  and  the  Angle  Opposite  One 
of  them  (a,  6,  a).  —  Find  /8  by 
the  sine  proportion, 


sin  yQ  =  sin  6 


Fig.  120. 

Find  c  by  (4)  and  (5),  Art.  147, 

.       1  ^      \r         ^^  sinl(«  +  ^) 

tan  I  c  —  tan \(.ci  —  o)  — — f— ^r-, 

2  2  V''  gin  1  («  —  ^) 

7  N  cos  i  (a  +  /3) 

tan  *  tf  =  tan  l  (a  +  6) f^ -^• 

2  2  V  ^  cos  I  (^a  —  13) 

Find  7  by  (6)  and  (7),  Art.  147, 
cot  J  7  =  tan  ^  (a 

cot  J  7  =  tan  -|-  («  +  /3) 


sing 
sin  a 


sinl(a  +  0 
'^^sinKa-fty 

cos|^(a  +  6) 


*) 


(1) 

(2) 
(3) 

(4) 
(5) 


cos^(a 

The  agreement  of  the  values  of  Jc  and  of  ^y  is  a  check 
upon  the  computation. 

Since  ^  is  found  by  means  of  its  sine,  it  may  be  either  in 

the  first  or  in  the  second  quad- 
rant ;  hence  there  may  be  two 
solutions.  If  b  differs  more  than 
a  from  90°,  y8  must  be  of  the 
same  species  as  5,  and  the  quad- 
rant in  which  /3  lies  is  fixed. 
But  if  b  does  not  differ  more 
than  a  from  90°,  we  cannot  de- 
termine by  the  first  rule  for 
species  the  quadrant  in  which  yS 
must  lie,  and  both  values  of  yS 
may  be  admissible.  Hence,  in- 
spect for  two  solutions  when  the  side  opposite  the  required  angle 
differs  less  from  90°  than  the  side  opposite  the  given  angle.  After 
finding  yS,  the  second  rule  of  Art.  146  will  show  whether  both 
values  are  admissible. 


Fig.  121. 


1.    Solve  the  triangle  when    a  =  148°34'.4,  6  =  142°  11'. 6,  a 
Since  6  differs  less  than  a  from  90°,  there  may  be  two  solutions. 


153°  17'.6. 


OBLIQUE   SPHERICAL   TRIANGLES. 


163 


logsin6  =  9.78746 
log  sin  0  =  0.65265 
col  sin  a  =  0.28282 
log  sin  /3 


i(«+6)=145°23'.0 
i(a  +  i3)=    92°35'.65 
i(a-/3)=    60''41'.95 


Ka-6)=      3°11'.4 
i(a  +  /3')=150°41'.95 
i(o-/5')=     2°36'.66 


9.72293 
31'^53'.7 


/3'  =  148°6^3 


The  second  rule  for  species  is  satisfied  for  both  /3  and  /S';  hence  there  are 
two  solutions. 

First.  Second. 

log  tan  i  (a  -  &)  =  8. 74612          or  8. 74612 

log  sin  Ha  +  /3)  =  9.99955  9.68966 

col  sin  ^a  -  i8)  =  0.05945  1 .34427 

logtan^c  =  8.80512 

^c=3°39M8 

c=  7°18'.36 

log  tan  K«  +  ^)=  9.83903  n       or 
log  cos  ^  (a  +  /3)  =  8.65573  n 
col  cos  1  (o  -  /3)  =  0.31034 
logtan|c  =  8.80510 
^c  =  3"39'.17 
c=7°18^34 

log  tan  |(o  -  /3)  =  0. 25089  or 

logsin|(a+ 6)=  9.75441 
col  sin  ^  (a  -  6)  =  1.25456 
logcoti7  =  1.25986 
^7  =  3°   8'.79 
7  =  6^22158 

log  tan  ^  (a  +  /3)  =  1.34383  n       or 
log  cos  |(a  +  &)  =  9.91538  n 
col  cos  ^la-b)  =  0.00067 
log  cot  ^7  =  1.25988 
|7  =  3^   8'.78 
7  =  6°17^56 

2.  a  =  40°  20'.4,  b  =  20°  18'.2,  a  =  60°  44'.4. 

.-.  /3  =  27°52'.9;   |c  =  23°34'.34  ;   i7  =  49°26'.7. 

3.  a  =  98°  16',  6  =  74°  38',    o  =  78°  40'. 

.-.  /3  =  72°49'.25;  ^c  =  75°53'.0  or  75°52'.6*  ;  ^7  =  76°1'.5  or  76°l'.l.* 

155.    Case  V.     Second   Method.     Given  a,  b,  a,  to  find  One 
Element  only. 

(1)    To  find  fi  only. 

sin/3=4^sin«.  (1) 

sin  a 

♦  These  values  would  be  taken,  since  a  small  error  in  /3  will  affect  them  less 
than  If  they  had  been  computed  from  the  other  formulas. 


9.78006 

31°  4'.  46 

c'  =  62°8'.92 

9.83903 n 

9.94055 n 

0.00045 

9.78003 

31°4'.39 

c'=62°8'.78 

8.65617 

9.75441 

1.25456 

9.66514 

65°  10'.68 

7'=130°21'.36 

9.74911  n 

9.91538  n 

0.00067 

9.66516 

65°  10'.62 

7'=130°21'.24 

164 


SPHERICAL   TRIGONOMETRY. 


(2) 


(3) 


(2)  To  find  c  only. 

cos  h  cos  c  -f-  sin  h  sin  c  cos  a  =  cos  a. 
Let  m  sin  M  =  sin  h  cos  a,  ] 

TwcosiHf  =  cos  5  j 

.  •.  m  cos  (iHf  —  c?)  =  cos  a. 

rn/r         \       COS« 

.  •.  COS  (M  —  c)  = 

m 

iff—  c  may  be  either  in  the  first  and  fourth  quadrants  or  in  the 

second  and  third  ;   if  there  are  two  solutions  both  values  of 

M-c  will  give  tf<180°. 

(3)  To  find  7  only. —From  (7),  Art.  124, 

cos  b  cos  7  +  sin  7  cot  a  =  cot  a  sin  h. 
.  *.  cos  b  sin  a  cos  7  +  sin  7  cos  a  =  cot  a  sin  5  sin  a. 
m  sin  i^=  cos  5  sin  a, 
wcos  Jf  =  cos  a. 
w  sin  (il[f -t- 7)  =  cot  a  sin  6  sin  «. 
cot  a  sin  5  sin  a 


Let 


(-t) 


.-.  sin(7l[f +7)  = 


m 


(5) 


3/4-7  i^^y  be  either  in  the  first  and  second  quadrants  or  in 
the  third  and  fourth  ;  if  there  are  two  solutions,  both  values  of 
M-\-y  will  give  7  <  180°. 

1.    a  =  148°34'.4,  6  =  142°11'.6,  a  =  153°17'.6. 
To  find  c. 

logsin&  =  9.78746  (1) 

log  cos  a  =  9.95101  n  (3) 

log  (m  sin  M )  =  9. 73847  n  (4) 

logsin  J/ =  9.75561  w  (7) 

log  cos  i«'=  9.91481  w  (8)     - 

log  cos  b  =  9.89767  »  (2) 

logtan  J/ =:  9.84080  (5) 

31  =  214°4.S^6  (6) 

cologm  =  0.01714  (9) 

log  cos  a  =  9.93111  n  (10) 

,  log  c<3S  (M  -c)  =  9.94825.  n  (11) 

J/-c  =  152°34'.8  (12) 

J/=214°43'.6  (14)             ]ogsin(iM"+ 7)=  9.67118  n    (12) 

M-  c'  =  207°25^2  (13)                            ilf+ 7  =  207°  58'. 2    (13) 

.:  c=    62°   8'.8  (15)                                    Jf=201°40'.6    (15) 

and  c'  =      7°18'.4  (16)                           i«'+ 7' =  332^JA8    (14) 

Two  values.  •••  7=     6°17'.6    (16) 

•  and  7^  =  130°  2V.2    (17) 

Two  value?.. 


To  find  7. 

log  cos  6  =  9.89767  n 

(1) 

log  sin  a  =9.65265 

(3) 

log  (m  sin  M)=  9.55032  71 

(5) 

logsinitf  =  9.56746  w 

(8) 

log  cos  3/=  9.96815  7i 

(9) 

log  cos  a  =  9.95101  n 

(4) 

logtani¥=  9.59931 

(6) 

iW^=20r40'.6 

(7) 

cologw  =  0.01714 

(10) 

logcota  =  0.21.393  n 

01) 

logsin?)=  9.78746 

(2) 

log  sin  0  =  9.65265 

(3) 

OBLIQUE  SPHERICAL  TRIANGLES.  165 

2.    a=40°20'.4,  6  =  20°  18'.2,  a  =  60°44'.4. 

.-.  3/=10°15'.0,  c  =  47°8'.7;    Jf  =  59°8'.8,  'y  =  98°53'.5. 
8.  a  =  98°  IC,  b  =  74°  38',  a  =  78°  40'. 

.-.  ilf=36°34'.0,  c  =  15r45'.4;    ilf  =  62°53'.9,  7  =  152°2'.6. 

156.  Case  VI.  Given  Two  Angles  and  the  Side  Opposite  One 
of  them  (a,  p,  a). — Find  b  by  the  sine  proportion, 

T        •     o  sin  a  >.^  ^ 

sin  0  =  sin  yS (1) 

sin  a 

Find  c  by  (4)  and  (5),  Art.  147, 

tan  J  c  =  tan  K«  -  ^)  "^^  f  ^^^  +  ^>  (2) 

tan  J  ^  =  tan  J  («  +  *)  ^^'f^^  +  ^>  (3) 

2  2  V  ^^  COS  ^  («  —  yS)  ""  ^ 

Find  7  by  (6)  and  (7),  Art.  147, 

cot  1 7  =  tan  1  (a  -  /3)  ^^^f^^  +  ^>  (4) 

2'  2v        ^^sinj(a  — 6)  ^^ 

^1  ^  1^         ,      ON  cos  A  (a   +^)  ,rx 

coti7-tanK«  +  ^)^^3|^^_^j-  (5) 

The  agreement  of  the  values  of  J  <?  and  of  J  7  is  a  check 
upon  the  computation. 

Since  b  is  found  by  means  of  its  sine,  it  may  be  either  in 
the  first  or  in  the  second  quadrant;  hence  there  may  be  two 
solutions.  If  /3  differs  more  than  «  from  90°,  ff  and  b  must  be 
of  the  same  species,  and  the  quadrant  in  which  b  lies  is  fixed. 
But  if  ff  does  not  differ  more  than  a  from  90°,  we  cannot  deter- 
mine by  the  first  rule  for  species  the  quadrant  in  which  b  must 
lie,  and  both  values  of  b  may  be  admissible.  Hence,  inspect  for 
two  solutions  when  the  angle  opposite  the  required  side  differs  less 
from  90°  than  the  angle  opposite  the  given  side.  After  finding  5, 
the  second  rule  of  Art.  146  will  show  whether  both  values  are 
admissible. 

1.  Solve  the  triangle  when  a  =  143°  17'.4,  jS  =  70°  18'.4,  a  =  160°40'.6. 
Since  /3  differs  less  than  a  from  90°,  there  may  be  two  solutions. 

logsin  a  =  9.51969  i  («  + /3)=  106°47'.9  ^(a-j8)=    36°29'.5 

log  sin  ^  =  9.97383  l(a  +  b)=    96°   2'.65  ^"'(rt  +  6')=  154°37'.95 

col  sin  g  =  0.22347  ^(a-6)=    64°37'.95  i(a-6')=      6°    2'.65 

log  sin  6  =  9.71699 
6=    31°24'.7 
6'  =  148°35'.3 


166 


SPHERICAL   TRIGONOMETRY. 


The  second  rule  for  species  is  satisfied  for  both  b  and  h' ;  hence  there  are 


two  solutions. 


First. 

log  tan  ^  (a  -  6)  =  0.32409 
log  sin  i  (o  +  /3)=  9.98106 
colsin  ^  (a -/3)  =  0.22570 

log  tan^c  =  0.53085 

lc=    73°35'.28 
c  =  147°  10'.  56 


log  tan  ^  (a  +  6)  =  0.97517  n 
log  cos  ^  (o  +  /S)  =  9.46091  n 
col  cos  i  (o  -  /3)  =  0.09478 

logtan^c  =  0.53086 

lc=    73°35'.30 
c  =  147°  lO'.OO 


Second. 
9.02483 
9.98106 
0.22570 

9.23159 
9°  40'.  38 
19^20'.76 

9.67591  n 
9.46091  n 
0.09478 


9.23160 
9°  40'.  39 
c'=19°20'.78 


log  tan  ^  (a  -  /3)  =  9.86908 
log  sin  J  (a  +  6)  =  9.99758 
col  sin  i  (a  -  6)  =  0.04403 

log  cot  ^7  =  9.91069 
.    ^7=    50°  51'.  00 
7  =  101°42'.00 

log  tan  ^  (a  -f  /3)  =  0.52016  n 
log  cos  ^  (a  +  6)  =  9.02241  n 
col  cos  ^  (a -6)  =  0.36813 

log  cot  ^7  =  9.91070 

^7=    50°50'.96 
7  =  101°41'.92 


9.86908 
9.63187 
0.97759 


0.47854 
18°  22'.74 
^'  =  36°45'.48 


0.52016  n 
9.95597  n 
0.00242 


0.47855 
18°  22'.71 
7'  =  36°45'.42 


2.  a  =  117°54'.4,  /3  =  45°8'.6,  a  =  76°37'.5. 


6  =  51°17'.9;  ^ c  =  20° 32'.3  or  20° 32'.4  ;  ^7  =  18°19'.4. 


8.  a  =  104°  40'.0,  iS  =  80°  13'.6,  a  =  126°  50'.4. 
.-.  b=   54°36'.8;  |c=73°48'.4  or  73°48'.5;  ^7=69°49'.5  or 
and   6'  =  125°23'.2;  ^c'=  3°25'.6  or  3°25'.5;  17'=  4°  8'.8. 


49'.6  ; 


157.    Case  VI.     Second  Method.     Given  a,  p,  a,  to  find  One 
Element  only. 

(1)  To  find  b  only. 

1      sin  S  .  ^-1 V, 

sm  0  = —  sm  a.  (1) 

sin  a 

(2)  To  find  c  owZ?/.— Permuting  (3),  Art.  124, 

cot  a  sin  c  —  cos  c  cos  yS  =  sin  y8  cot  a. 
.  •.  cos  a  sin  <?  —  sin  a  cos  c  cos  yS  =  sin  a  sin  ^  cot  a. 


Let 


OBLIQUE   SPHERICAL   TRIANGLES. 

m  sin  M  =  sin  a  cos  yS,  ) 

mcosM=cosa.  j 

.  •.  m  sin  ((?  —  M)  =  sin  a  sin  /8  cot  a. 

•    ^         7i.r\      sin  a  sin  8  cot  a 

.-.  sin  ((?  — i)[f)  = ^ 

m 


167 

(2) 

(3) 


c  —  M  may  be  either  in  the  first  and  second  quadrants,  or  in 
the  third  and  fourth ;  if  there  are  two  solutions,  both  values  of 
c-M\vi\\  give  c<180°. 

(3)   To  find  7  onli/, 

—  cos  yS  cos  7  +  sin  jS  sin  7  cos  a  =  cos  a. 
Let  m  smM=  cos  a  sin  j3,  | 

mcosiHf  =  cosyS.  J 

. '.  m  cos  (iHf  H-  7)  =  —  cos  a. 
cos  a 


(4) 


cos(M-{-  7)=  — 


m 


(5) 


ilf  +  7  may  be  either  in  the  first  and  fourth  quadrants,  or  in 
the  second  and  third ;  if  there  are  two  solutions,  both  values  of 
if  +  7  will  give  7  <  180°. 


:  =  143°  17'. 4,  /3 
To  find  c. 
log  sin  a  =  9.51969 
logcosiS  =  9.52761 


log  (wi  sin  ilf)=  9.04730 
log  sin  3/ =  9.06947 
log  cos  iW=  9.99699  w 
log  cos  a  =  9.97482  n 

log  tan  i¥=  9.07248  n 
Jf=173°15'.7 


(1) 
(3) 

(5) 
(8) 
(9) 
(2) 

(6) 
(7) 

cologm  =  0.02217  (10) 

logsina  =  9.51969  (1) 

log  sin /3  =  9.97383  (4) 

log  cot  0  =  0. 12746  w  (11) 

log  sin  {c-  M)=  9.64315  n  (12) 

c-M.=  20Q°   5M  (13) 

Jf=173°15'.7  (15) 

c' -  i»f  =  333°  54'.9  (14) 

c  =    19°20'.8    (16) 
c'  =  147°10'.6    (17) 
Two  values. 


70°18'.4,  a  =  160°40'.6. 

To  find  7. 
log  cos  rt  =  9.97482  n 
log  sin  /3  =  9.97383 


log  (m  smM)=  9.94865  n 
log  sin  il/'=  9.97082  « 
logcositf=  9.54977 
log  cos  j8  =  9.52761 

log  tan  iJf=  0.42104  n 
M  =  290°  46'.2 


(1) 
(2) 

(4) 
(7) 
(8) 
(3) 

(5) 
(6) 

colog(- w)=  0.02217  w      (9) 
log  cos  a  =  9.90400  n    (10) 

logcos(ilf+ 7)=  9.92617  (11) 

M-\-y=   32°28'.2  (12) 

ilf=290°46'.2  (14) 

i»f+7'=327°31'.8  (13) 

7  =101°42'.0    (15) 
7'=    36°45'.6    (16) 
Two  values. 


168 


SPHERICAL   TRIGONOMETRY. 


2.    a  =  117°54'.4,  ^  =  45°8'.G, 


a 


'6''  37' 


31=71"  22'.3,  c  =  41°  4'.9;  M  =  13°  5'.3,  7  =  36°  38'.8. 
:80°13'.6,  a  =  126°50'.4. 
167°  14'.0,  c  =  147°  36'.9  or  6°  51'.1 : 


3.    a  =  104°40'.0,  )S 
.-.  31  = 


3f=~'JS°  58'.3,  7  =  139°  39'.0  or  8°  17'.6. 

OBLIQUE   TRIANGLES   SOLVED   BY   RIGHT  TRIANGLES. 

158.    General  Method.  —  From  any  vertex  O  of  the  triangle 
draw  an  arc  p  of  a  great  circle  perpendicular  to  the  opposite 


z)^-:- 


Fig.  122. 


side,  dividing  the  triangle  into  two  right  triangles.  Denote 
the  segments  of  the  side  by  m  and  n,  and  the  corresponding 
segments  of  the  angle  by  M  and  N. 

The  opposite  side  must  in  some  cases  be  produced  to  meet 
the  perpendicular  arc,  as  in  Fig.  123.  The  segments  of  the 
side  are  AD  and  DB,  and  their  signs  are  so  taken  that  tlieir 
algebraic  sum  shall  be  equal  to  the  side  ;  that  is,  if  a  segment  is 
entirely/  exterior  to  the  triangle,  it  is  negative. 

The  perpendicular  p  may  have  either  of  two  supplemental 
values ;  we  shall  always  place  it  in  the  same  quadrant  as  its 
opposite  angle  in  the  triangle  first  used  in  the  solution,  in 
accordance  with  the  rule  for  species. 


159.    Special  Formula.  —  To  prove 

tan  I  {tn  +  n)  tan  |  {m  -  n)  =  tan  |  (a  +  6)  tan  |  (a  -  6). 

In  both  Fig.  122  and  Fig.  123,  by  Napier's  rules, 
cos  a  =  cos  m  cosjo,     and     cos  b  =  cos  n  cos  p. 


(1) 


OBLIQUE  SPHERICAL   TRIANGLES.  169 

cos  a      cos  h 


•  * 

cosjt? 

= = 

cosm 

cosw 

cos  w 

cos  a 

cosw 

cos  6 

cosm  - 

-  cosw 

cos  a  - 

-  COS  6 

cos  m  +  cos  n  cos  a  +  cos  6' 
which  becomes,  from  (4)  and  (3),  Art.  73, 
tan  \(m  +  ii)  tan  J  (jn  —  n)  =  tan  J  («  +  ^)  tan  |  (a  —  5).   q.e.d. 

160.    Case  I.     Given  a,  6,  c.  —  From  (1),  Art.  159,  we  have 
tan  ^  (jn  —  n)=  tan  ^  (a  -\-  b^  tan  |^  (a  —  ft)  cot  ^  e,        (1) 

since  m  +n  =  c.  We  shall  consider  |^  (ra  —  n)  as  being  numeri- 
cally less  than  90°,  so  that  it  will  be  a  negative  angle  when  its 
tangent  is  negative.     After  |  (m  —  n)  has  been  found,  we  have 


n  =  ^c  —  l.  (jn  —  n) 

A  negative  value  of  m  ov  oi  n  indicates  that  the  segment,  and 
hence  the  corresponding  triangle,  is  exterior  to  the  given  tri- 
angle. Note  that  m  is  always  measured  from  the  side  that  is 
called  «,  and  n  from  h. 

In  the  triangles  A  CD  and  I)  CB  we  now  know  the  two  sides, 
so  that  the  other  elements  can  be  found  by  Napier's  rules. 
The  example  shows  the  method  of  finding  the  elements  of  the 
original  triangle  from  the  results  of  the  computation. 

1.  a  =  114°  43'.3,  h  =  136°  lO'.G,  c  =  43°  18'.5. 

From  (1),  ^  (m  -  n)=  33°oG'.81,  whence  m  =  55°36'.06,  w  =-  12°  17'.56. 
The  negative  value  of  n  shows  that  ACD  is  exterior  to  the  triangle. 

From  BCD  we  find  DB€  =  ^  =  132°  15'.3,  DCB  =  iJf  =  65°  17'. 0. 

From  ^CZ>  we  find  D.4C=  180°- «=  103°  11'.6,  ^CZ)  =  iV=  -  17°  57'.5, 
giving  iY  the  negative  sign  since  it  is  exterior  to  the  triangle.     Hence 

a  =  76°  48'.4  ;   7  =  3/-I- iV=  47°  19'.5. 

2.  a  =  76°40'.4,  5  =  54°21'.3,  c  =  36°8'.7. 

.-.  J(»w-w)=53°0'.38:  w  =  71°4'.73:  n  =  -  34°  66'.03  ;  /3  =  46°17'.3; 
Jlf=76°27'.0;  iV  =  -  44°  48'.2  :  7  =  31°38'.8;  a  =  120°3'.6. 


170  SPHERICAL   TRIGONOMETRY. 

3.  a  =  124°  34'.0,  h  =  6Q°  7'.2,  c  =  109°  43'.5. 

.-.  J(m-w)  =  -76°37'.32;  m  =  -  21°45'.57  ;  n  =  +  131°29'.07  ;  /3  =  74°  1'.7  ; 
ilf  =  -  26°  45'.6  ;  a  =  120°  2'.7  ;  iV  =  +  124°  69'.2  ;  7  =  98°  13'.6. 

4.  a  =  30°17'.6,   6=22°14'.4,  c  =  18°51'.8. 

.-.  ?>i  =  21°  14'.6  ;  n  =  ~2°22'.8;  /3  =  48°17M;  J/=45°54'.8; 
a  =  95°  50'.0  ;  iV^  =  -  6°  17'.9  ;  7  =  39°  36'.9. 

5.  a  =  130°46'.0,  b  =  113°21'.4,  c  =  102°  16'.2. 

.  •.  i  (m  -  «)  =  -  11°  8'.6  ;  m  =  39°  59'.5  ;  n  =  62°  16'.7  ;  /3  =  136°  19'.25  ; 
Jf  =  58°  3'.4  ;  a  =  145°  15'.9  ;  i\r  =  74°  37'.75  ;  7  =  132°  41'.2. 

161.  Case  II.  Given  a,  p,  y,  —  ^Pply  the  method  of  Case  I 
to  the  polar  triangle,  and  thence  find  the  elements  of  the 
original  triangle. 

1.  a  =  116°  19'.4,  /3  =  83°  19'.2,  7  =  106°  10'.6. 
In  the  polar  triangle, 

a'  =  63°  40'.6,  b'  =  96°  40'.8,  c'  =  73°  49'.4. 
.-.  J  (m'  -  n')  =  -  66°  18M,  m'  =-  29°  23'.4,  n'  =  +  103°  12'.8. 
The  negative  value  of  m'  shows  that  B'C'D'  is  exterior  to  the  triangle. 
From  B'C'D'  we  find 

D'B'C  =  180°  -  /3'  =  73°  49'.2,  D'C'B'  =  3I'=-  33°  11'.8, 
giving  M'  the  negative  sign  since  it  is  exterior  to  the  triangle. 
From  A' CD'  we  find 

D'A'C  =  a'  =  60°  4'.7,  N'  =  +  101°  25'. 5. 
.  •.  p'  =  106°  lO'.S,  7'  =  M'  +  N'=  68°  13'.7. 
Passing  from  the  polar  to  the  original  triangle, 

a  =  119°  55'.3  ;  6  =  73°  49'.2  ;  c  =  111°  46'.3. 

2.  a  =  110°  36'.4,  /3  =  122°  8'.7,  7  =  140°  20'.3. 

.  •.  J  (m'  -  n')  =  29°  27'.90  ;  m'  =  49°  17'. 75  ;  w'  =  -  9°  38'.05  ; 
/3'  =  64°  4'.9  ;  M'  =  54°  5'.4  ;  a'  =  96°  7'.4  ;  iY'  =  -  11°  24'.0  ;  7'  =  42°  41'.4  ; 
.-.  a  =  83°  52'.6,  b  =  115°  55M,  c  =  137°  18'.6. 

3.  o  =  120°  50'.6,  /3  =  78°  6'.1,  7  =  81°  12'.3. 

.  •.  i  (m'  -  w')  =  -  63°  33'.19  ;  m'  =  -  14°  9'.34  ;  n'  =  112°  67'.04 ; 
/3'  =  98°  39'. 7  ;  M'=-  1G°  33'.0  ;  a>  =  60°  9'.6  ; 
N'  =  109°  46'.0 ;  7'  =  93°  IS'.O  ; 
.:  a  =  119°  50'.4,  b  =  81°  20'.3,  c  =  86°  47'.0. 

4.  a  =  80°  20'.2,  /3  =  73°  46'.7,  7  =  54°  8'.5. 

.-.  i  (m'  -  n')  =  7°  15'.69  ;  m'  =  70°  11'.44  ;  n'  =  55°  40'.06  ; 
/3'  =  118°  12'.7  ;  M'  =  72°  37'.5  ;  a'  =  115°  12'.8  ; 
N'  =59°19'.l;  7' =  131°  56'.6  ; 
.'.  a=  64°  47'.2,  b  =  61°  47'.3,  c  =  48°  3'.4. 


OBLIQUE   SPHERICAL   TRIANGLES. 


171 


6.  a  =  100°  61'.3,  /3  =  80°  47'.C,  7  =  74°  3'.3. 

.  •.  J  (m'  -  n')  =  -  83°  60'.76  ;  m'  =  -  30°  62'.41  ;  n'  =  130°  49'.11  ; 
/3'  =  96°  35'.0  ;  3P  =  -  31°  30'.0  ;  a'  =  81°  16M  ; 
N'  =  136°  6'.8  J  7'  =  104°  36'.8 ; 
.:  a  =  98°  44'.9,  b  =  83°  25'.0,  c  =  76°  23'.2. 

162.  Case  III.    Given  a,  6,  -y.  —  From  the  end  of  one  of  the 

sides,  as  5,  let  fall  an  arc  of  a  great  circle  perpendicular  to  the 

other  side.     In  the  triangle  J) AC  we 

know  b  and  7 ;    hence  we  find  w,  iV, 

and  p  by  Napier's  rules,  considering 

p  as  of  the  same  species  as  7.     Then 

m  =  a  —  n,  being  negative  when  n>a, 

showing  that  the  triangle  BAD  is  then 

exterior  to  the  triangle  BA  C. 

Now  in  the  triangle  BAD  we  know 
DB  and  AD^  and  we  find  <?,  iHf,  and 
ABD  by  Napier's  r«les. 

1.  a  =  105°  14'.8,  6  =  43°  17'.2,  7  =  112°  47'.4. 

.;.   n  =  159°  57'.3,  N=  150°  0'.4,  p  =  140°  47'.53. 
. '.  w  =  —  54°  42'.5,  showing  that  BAD  is  exterior  to  BAC. 
In  the  triangle  BAD  we  find 

ABD  =  180°  -  j8  =  135°  0'.8,  c  =  116°  35'. 6,  ilf  =  -65°  63'.7, 
giving  M  the  negative  sign  since  it  is  exterior  to  the  triangle. 
Hence  ^  =  44°  59'.2,  a  =  iV^+ ilf=  84°  6'. 7. 

2.  a  =  103°  44'.7,  b  =  64°  12'.3,  7  =  98°  33'.8. 

.-.  iY=100°54'.7;  ?i  =  162°  52'.6  ;  ;)=  117°  5'.1 ;  m=-69°7'.9; 
c  =  103°  30'.6  ;  ^  =  m°  18'.0  ;  31  =  -  61°  58'. 7  ;  a  =  98°  56'.0. 

3.  a  =  156°  12'.2,  b  =  112°  48'.6,  7  =  76°  32'.4. 

.:  3f=  148°  18'.6  ;  n  =  151°  2'.3  ;  p  =  63°  41'.8  ;  ?»  =  5°  9'.9  ; 
c  =  63"  48'.8  ;  /3  =  87°  27'.1 ;  M  =  5°  45'.5  ;  a  =  154°  4M. 

163.  Case  IV.  Given  a,  p,  c.  —  Let  fall  from  the  vertex  of 
one  of  the  angles,  as  a  =  BAC  (Fig.  124),  an  arc  of  a  great 
circle  perpendicular  to  the  opposite  side.  In  the  triangle  ABD 
we  know  e  and  yS,  and  we  find  m,  M,  and  p  by  Napier's  rules, 
considering  p  as  of  the  same  species  as  ^.  Then  iV  =  a  —  iltf,  a 
negative  value  of  JS^  showing  that  the  point  D  lies  on  BC  pro- 
duced, the  triangle  A  CD  being  then  exterior  to  the  given 
triangle. 


172  SPHERICAL   TRIGONOMETRY. 

In  the  triangle  ACD  we  now  know  p  and  CAD^  and  we 
find  6,  7,  and  n  by  Napier's  rules. 

\.    a  =  140°  43'.2,  ^  =  100°  4'.6,  c  =  60°  43'.6. 

.  ,  rr>  =  162°  39'.9,  j?  =  120°  48'.86,  31  =:.  160°  VJ. 
Then  iV  =  a  -  i¥  =  -  19°  18^5. 

.  •.  &  =  119°  22'.5,  ACD  =  180°  -  7  =  99°  45M,  n  =  -  16°  44'.8, 
giving  n  the  negative  sign  since  it  is  exterior  to  the  triangle. 

.-.  7  =  80°  14'.9,  a  =  m-\-n  =  145°  55M. 

2.    a  =  104°  30'.7,  /3  =  62°  52M,  c  =  56°  6'.4. 

.  •.  M  =  42°  34'.8  ;  iNr=  61°  55'.9  ;  m  =  34°  10'.2  ;  p  =  47°  37'. 5  ; 
b  =  66°  46'.0  ;  7  =  53°30'.4  ;  7i  =  54°  10'.7  ;  a  =  88°  20'.9. 
8.    a  =  140°  24'.6,  /3  =  12°  18'.6,  c  =  28°  7'.7. 

. •.  ilf  =  79°  6'.4  ;  iNT  =  61°  18'.2  ;  w  =  27°  34'.7  ;  p  =  5°  46M  ; 
&  =  11°  52'.9  ;  7  =  29°  13'.3  ;  n  =  10°  24'.3  ;  a  =  37°  59'.0. 

> 

164.    Case  V.    Given   a,  b,  a.  —  Let  fall  an  arc  of  a  great 
circle  from  the  intersection  of  a  and  J, 
^^  perpendicular  to  c.     In  this  case  there 

^^^^Vi\  will   be    two  solutions   if  a  is  inter- 

y^        /    1     \a       mediate  in  value  between  p  and  both 
X  7      'f    \       b  and  180°  -  h  (Art.  120). 

''^^O-— -^/         \d     ^r         ^^^  ^^®  triangle  ACD,  knowing  b 
"'^'^  ^  and  «,  find  771,  M,  and  p  by  Napier's 

rules.  Then  in  the  triangle  DOB^ 
knowing  p  and  a,  find  DB^  DCB^  and  BBC.  Then  in  the  tri- 
angle A  CB  we  have 

c  =  AB  =  m-\-BB,  y  =  ACB  =  lM-\- BOB,  ^  =  BBO; 
and  in  the  triangle  ACB' ^ 
c'=AB'  =  m-BB,  y' =ACB' =  M-BOB,  j3' =  180'' -BBC. 

1.    a  =  148°  34'.4,  h  =  142°  11'.6,  a  =  153°  17'.6. 

.-.  p  =  164°  0'.52,  and  there  are  two  solutions. 
->.  =  34°  43'.5,  M  =  68°  19'.4. 
Also,        DB  =  27°  25M,  DBC  =  148°  6'.3,  DCB  =  62°  1'.8. 
.-.  c  =  62°  8'.6,  7  =  130°  21'.2,  ^  =  148°  6'.3, 
and  c'  =  7°  18'.4,  7'  =  6°  17'.6,  j8'  =  31°  53'.7. 


OBLIQUE   SPHERICAL  TRIANGLES.  173 

2.  a  =  40°  20'.4,  b  =  20°  18'.2,  a  =  60°  44'.4. 

.'.  p  =  17°  37'.3  ;  m  =  10°  IG'.O  ;  ^  =  30°  61 '.2  ;  /3  =  27°  52'.9  ; 
DB  -  36°  53  .7  ;  DCB  =  68°  2'.3  ;  c  =  47°  8'.7  ;  7  =  98°  63'.6. 

3.  a  =  98°  IC,  &  =  74°  38',  a  =  78°  40'. 

.:  p  =  70°  59'.25  ;  m  =  35°  34'.0  ;  iW=  37°  6'.1  ;  /3  =  72°  49'.25  ; 
DB  =  116°  11'.4  ;  DCB  =  114°  66'.4  ;  c  =  151°  45'.4  ;  7  =  152°  2'.5. 

165.  Case  VI.  Given  a,  p,  a.  —  Pass  to  the  polar  triangle, 
in  which  we  shall  know  a',  Z>^  and  a',  and  solve  by  the  method 
of  Art.  164.  There  may  be  two  solutions  of  the  polar  triangle, 
and  therefore  of  the  triangle  itself. 

1.  a  =  143°  17'.4,  /3  =  70°  18'.4,  a  =  160°  40'.6. 

.-.  a'  =  36°  42'.6,  h>  =  109°  41'.6,  a'  =  19°  19'.4. 
•  •.  p'  =z  18°  9'.  13,  and  there  will  be  two  solutions. 
M'  =  96°  44'.3,  m'  =  110°  46'.3. 
Also       D'B'  =  32°  28'.25,  D'C'B'  =  63°  54'.9,  D'B'C  =  31°  24'.7. 
.'.  ci'  =  143°  14'.55,  7i'  =  160°  39'.2,  /3i'  =  31°  24'.7, 
and  ci"  =  78°  18'.05,  71"  =  32°  49'.4,  jSi"  =  148°  35'.3. 

Taking  the  supplements  to  obtain  the  elements  of  the  original  triangle, 
7  =  36°  45'.45,  c  =  19°  20'.8,  b  =  148°  35'.3, 
and  7'  =  101°  41'.95,  c'  =  147°  10'.6,  b'  =  31°  24'.7. 

2.  o  =  117°  54'.4,  p  =  45°  8'.0,  a  =  76°  37'.5. 

.:  p'  =  136°  23'.8  ;  31'  =  18°  37'.7  ;  m'  =  13°  5'.3  ; 
D'C'B'  =  120°  17'.5  ;  D'B'C  =  128°  42'.1  ;  D'B'  =  130°  15'.9  ; 
7'  =  138°  55'.2  ;  c'  =  143°  21'.2  ; 
.  •.  6  =  51°  17'.9  ;  c  =  41°  4'.8  ;  7  =  36°  38'.8. 

3.  a  =  104°  40'.0,  /3  =  80°  13'.6,  a  =  126°  50'.4. 

.-.  p'  =  52°  3'.8  ;  31'  =  102°  46'.0  ;  m'  =  106°  1'.7  ; 
D'C'B'  =  70°  22'. 9  ;  D'B'C  =  54°  36'.8  ;  D'B'  =  65°  40'.7  ; 
7i'  =  172°  8'.0  ;  71"  =  32°  23'.1  ;  d'  =  171°  42'.4  ; 
ci"  =  40°  21'.0  ;  /3i'  =  54°  36'.8  ;  /3i"  =  125°  23'.2. 
.  ••  c  =  7°  51'.1 ;  7  =  8°  17'.6  ;  b  =  125°  23'.2  ; 
and         c'  =  147°  36'.9 ;  7'  =  139°  39'.0  ;  h'  =  54°  36'. 8. 


CHAPTER  XII. 


APPLICATIONS   OF   SPHERICAL   TRIGONOMETRY. 


166.  To  find  the  Shortest  Distance  between  Two  Points  on 
the  Surface  of  the  Earth,*  the  earth  being  treated  as  a  sphere.  — 
North  latitudes  and  west  longitudes  are  considered  positive. 
Let  QQ'  be  the  equator,  P  the  north  pole,  A  and  B  the  two 

points,  and  PM  the  meridian 
from  which  the  longitudes  are 
measured.  The  longitude  of 
A  is  MPC  and  that  of  B  is 
MPD^  both  being  positive 
since  they  are  measured  west- 
ward. The  latitudes  are  CA 
and  DB^  the  former  being 
negative  since  it  is  measured 
southward. 

In  the  triangle  APB  the 
sides  AP  and  BP  are  found 
by  algebraically  subtracting 
the  latitudes  from  90°,  and  the  angle  APB  is  the  algebraic 
difference  of  the  longitudes.  Hence  we  know  two  sides  and 
their  included  angle,  so  that  we  can  solve  the  triangle,  using 
the  method  of  Art.  151  when  the  distance  only  is  required, 
and  that  of  Art.  150  when  we  wish  to  find  all  the  elements. 

.  1.   Find  the  shortest  distance  between  New  York,  40°  45'.4  N.,  73°  58'. 4  W., 
and  Rio  Janeiro,  22°  54'.4  S.,  43°  10'.4  W. 

.-.  I?P=49°  14'.6,  ^P=112°  54'.4,  ^PS=30°  48'.0.     Ans.  .45=69°  48'. 2. 

2.  Find  the  shortest  distance  between  New  York,  40°  45'.  4  N.,  73°  58'.4  W., 
and  Paris,  48°  50'.2  N.,  2°.20'.2  E.  Ans.  AB  =  52°  2C'.8. 

*  The  shortest  distance  between  two  points  on  a  sphere  is  the  arc  of  the 
great  circle  passing  through  the  points. 

174 


Fig.  126. 


APPLICATIONS  OF   SPHERICAL   TRIGONOMETRY.       175 

If  the  bearings  of  the  great  circle  AB  at  A  and  B  are  required,  it  -will  be 
necessary  to  find  the  angles  PAB  and  PBA. 

3.  A  ship  sailed  from  Calcutta,  22°  34'.8  N.,  88°27'.3E.,  on  an  arc  of  a 
great  circle  to  Melbourne,  37°  48'.0  S.,  144°  58'.0  E.  Find  the  distance  sailed 
and  the  bearings  *  at  both  points. 

Ans.  At  Calcutta,  S.  41°  56'. 61  E. ;  at  Melbourne,  S.  61°  21 '.47  E. ;  dis- 
tance, 80°22'.4or80°22'.6. 

4.  A  ship  sailed  from  the  Cape  of  Good  Hope,  34°  22'  S.,  18°  29'  E.,  on  an 
arc  of  a  great  circle  to  Cape  St.  Roque,  5°  28'  S.,  36°  16'  W.  Find  the  distance 
sailed  and  the  bearings  *  at  both  points. 

Ans.  At  G.  H.,  N.  72°  28'.0  W.  ;  at  S.  R.,  N.  52°  16'.0  W. ;  distance,  57°  20'.4. 

6.  A  ship  sailed  from  Bombay,  I&°  56'  N.,  72°  53'  E.,  on  an  arc  of  a  great 
circle  to  the  Cape  of  Good  Hope,  34°  22'  S.,  18°  29'  E.  Find  the  distance  sailed 
and  the  bearings  *  at  both  points. 

Ans.  At  Bombay,  S.  44°  12'.8  W. ;  at  G.  H.,  S.  53°  2'.6  W. ;  distance, 
74°  15'.2  or  74°  15'.4. 

6.  A  ship  sailed  from  Bombay,  18°  66'  N.,  72°  53'  E.,  on  an  arc  of  a  great 
circle  for  the  Cape  of  Good  Hope,  34°  22'  S.,  18°  29'  E.  Find  the  distance  to  the 
equator  and  the  bearing*  and  longitude  at  the  equator.  [Use  the  triangle 
BDE;  the  angle  PBA  -  135°  47'.2  was  found  in  Ex.  6.] 

Ans.  S.  41°  16'.  1  W. ;  distance,  25°  34'.5  ;  longitude,  65°  21'.8  E. 

7.  From  a  point  whose  latitude  is  17°  N.  and  longitude  130°  W.  a  ship 
sailed  an  arc  of  a  great  circle  over  a  distance  of  4150  miles,  starting  S.  64°  20'  W. 
Find  its  latitude  and  longitude  if  the  length  of  1°  is  69^  miles. 

Ans.  Lat.,  19°  40'.62  or  19°  40'.60  S. ;  Long.,  178°  20'.9  W. 

167.  Given  the  Lengths  of  the  Three  Edges  of  a  Parallelo- 
piped  that  meet  in  a  Point,  and  the  Angles  between  them,  to 
find  the  Surface  and  the  Volume  of  the  Parallelopiped.  —  Let 
OGr  he  the  solid,  AD  the  perpendicular  from  A  to  BOO,  and 
hence  AOD  a  plane  per- 
pendicular to  BOO.  Let 
the  angles  and  edges  be 

BOC=:a,  AOC=h, 
AOB  =  c,      OA  =  l, 


0B  = 


m, 


00  = 


n. 


Describe  a  sphere  with  a 
radius  of  unity  about  0  as  a 
center,  its  intersections  with 
the  planes  forming  the  figure  marked  by  the  primed  letters. 


Fig.  127. 


The  course  of  the  ship. 


176 


SPHERICAL   TRIGONOMETRY. 


Then  the  surface  is 

^=2  OBEC^-  2  OAFC+  2  OBHA 

=  2  {mn  sill  a  +  In  sin  b  +  Im  sin  c).  (1) 

In  the  triangle  A'B'B',  right-angled  at  D',  we  have 

sin  B'A'  =  sin  B'A'  sin  A'B'I)'  ; 

.  •.  sin  D'A'  =  sin  6>  sin  A'B'L'. 

But  in  the  triangle  A' B'  C  we  know  the  three  sides  a,  b^  c  ; 
hence 

sin  A'B'B'  =  2  sin  l  A'B'B'  cos  ^  J.'^'i>' 

2 


sin  a  sin  <? 
.-.  fiuiD'A'  =  sin  BOA 
2 


Vsiii  s  sin  (s  —  «)  sin  (s  —  ft)  sin  (.s-  —  c). 


sin  a 
i>A=  OA  sin  BOA 

21 


Vsin  s  sin  (s  —  «)  sin  (s  —  b)  sin  (s  —  <?). 


sin  a 

Hence  the  volume  is 
F=  OB  EC  X  D^ 


Vsin  s  sin  (s  —  a)  sin  (s  —  6)  sin  (s  —  <?). 


(2) 


=  2  Zww  Vsin  8  sin  (s  —  a)  sin  (s  —  ft)  sin  (s  —  c). 

168.  To  find  the  Volume  of  a  Regular  Polyhedron.  —  Let 
AB  be  the  edge  in  which  two  adjacent  faces  intersect,  B  its 
middle  point,  C  and  B  the  centers  of  the 
polygonal  faces,  and  0  the  center  of  the  sphere 
inscribed  in  the  polyhedron,  the  faces  being 
tangent  to  the  sjjhere  at  0  and  B.     Then 

BC=BB;  BA  =  BB; 

CDA  =  CBB  =  BBA  =  EBB  =  90°; 

i>(7O  =  90°;  BEO==90\ 

Let  a  =  length  of  an  edge  AB^ 

s  =  number   of  sides    of   each   polygonal 

face, 
n  =  number  of  faces  meeting  at  a  vertex 
of  the  polyhedron, 


APPLICATIONS  OF   SPHERICAL   TRIGONOMETRY.       177 

iV^=  number  of  faces  of  the  polyhedron, 
E  =  edge  angle  CDE  of  the  polyhedron. 

i   Then        CD  =  AD  Got  ACD  =  \a  cot  ^-^^' 

CO  =  CD  tan  CDO  =  CD  tan  -^  E, 

.'.   CO  =  I  acot^^  tan  IE,  (1) 

About  0  as  a  center,  with  a  unit  radius,  describe  a  sphere, 
and  let  its  intersections  with  the  three  planes  form  the  triangle 
A' CD'.     Then 

A'C'D'  =  ACD  =  ^^;    A'i>'(7'  =  90°;    C'^'D' =  lM2!. 
s  n 

By  Napier's  rules, 

cos  C'A'D'  =  cos  CD'  sin  A^  CD', 

180°  ^,j.f   .    180° 

or  cos =  cos  C^'-D' sin 

n  s 

But 

cos  CD'  =  cos  COD  =  cos  (90°  -  CDO}  =  sin  CD0=  sin  J  J57. 

180°        .    1  rr  .     180° 
.  •.   cos =  sm  J  E  sin 

•     1  XT            180°            180°  ,oN 

.  •.  sm  -^  E  —  cos cosec (2) 

7h  8 

Then,  if  A  is  the  area  of  a  face,  the  volume  is 

V=lCOxAxN=  ^\Ma^  cot2 1?^  tan  ^  E.         (3) 
Find  -1 -^from  (2)  and  then  Ffrom  (3). 

1.  Dodecahedron,  formed  by  12  regular  pentagons,  3  meeting  at  a  vertex. 

.  •.  s  =  5,  n  =  3,  iV  =  12.  log  cos  60°  =  9.G9897  lo^  ^  =  0  39-Q4 

log  cosec  36°  =  0.23078  ^24        "     '' 

log  sin  I  E  =  9.92975      ^'^S  cof^  36°  =  0.27748 

log  tan  ^E=z  0.20896 

0.88438 

.-.    F=  7.663  a*. 

2.  Tetrahedron,  formed  by  4  equilateral  triangles,  3  meeting  at  a  vertex. 

■.-.  s  =  3,  71  =  3,  iY=4.  Ans.   F=  0.1179 a^. 

3.  Cube,  formed  by  6  squares,  3  meeting  at  a  vertex. 

.-.  s  =  4.  n  =  3,  iV  =  6.  Ans.  V=a\ 

CROCK.  TRIG. — 12 


178 


SPHERICAL   TRIGONOMETRY. 


4.   Octahedron,  formed  by  8  equilateral  triangles,  4  meeting  at  a  vertex 
.-.  s  =  3,  «  =  4,  N=S.  Ans.  V=0A7Ua\ 

6.    Icosaliedron,  formed  by  20  equilateral  triangles,  5  meeting  at  a  vertex. 
.-.  s  =  3,  n  =  5,  iV=20.  Ans.    F=  2.182  a^. 

169.    If  from  Any  Point  in  a  Trirectangular  Triangle  Arcs  of 
Great  Circles  are  drawn  to  the  Vertices, 

COS'^  a  +  COS^  p  +  COS^  7  =  1» 

where  «,  /3,  and  7  are  the  arcs.  — In  Fig.  129,  produce  YP  and 
ZP  to  D  and  U.     In  the  right  triangle  PDX, 

sin  PD  =  sin  a  sin  PXD  ;    .-.   cos /3  =  sin  ct  sin  PXi>.    (1) 

In  the  right  triangle  PJSX, 

sinPU  =  sinasinPXU ;     .-.    cos  7  =  sin  a  cos  PXD.    (2) 

Squaring  (1)  and  (2),  and  adding,  we  have 

cos^  yS  +  cos^  7  =  sin^  «. 

.  •.  cos^  a  +  cos^  /3  +  cos^  7  =  1.  Q.E.i). 


Fig.  130. 

170.  If  from  Any  Two  Points  P  and  I*'  in  a  Trirectangular 
Triangle  Arcs  of  Great  Circles  are  drawn  to  the  Three  Vertices, 
and  if  v  is  the  Length  of  the  Arc  rr',  prove  that 

cos  V  =  cos  a  cos  a'  +  COS  p  cos  p'  +  cos  7  cos  y'. 

In  the  triangle  PYP^  (Fig.  130), 

cos  V  =  cos  yS  cos  /3^  +  sin  yQ  sin  yS'  cos  PYP'.  (1) 

But 

cos  PZP^  =  cos  CZYPf  -  ZYP^. 

.-.  cos P rP'  =  cos  ZrP^  cos  ZrP  + sin  ZZP' sin  ZrP.    (2) 


APPLICATIONS  OF   SPHERICAL   TRIGONOMETRY.       179 


(3) 


In  ZYP,    cos 7  =  sin y3  cos ZYF.* 
In  ZYF^,  cosy  =  sinks'  cos  ZYF'.* 
In  XYF,   cos  «  =  sin  /9  cos  XYF  *  =  sin  /3  sin  ZYF. 
In  XrP',  cos  a'  =  sin  /S'  cos  XYF'  *  =  sin  /3'  sin  ZYF'. 
Substituting  in  (1)  the  values  found  from  (2)  and  (3), 

cos  V  =  cos  /3  cos  fi'  +  cos  7  cos  y'  -\-  cos  a  cos  a',     q.e.d. 

This  is  the  formula  for  the  cosine  of  the  angle  between  two 
lines  in  space,  the  angles  made  by  them  with  tliree  lines  at  right 
angles  to  each  other  being  «,  yS,  7,  and  a'^  13',  7',  respectively. 

171.  To  find  the  Angle  a'  between  the  Chords  of  Two  Sides 
of  a  Spherical  Triangle,  having  given  the  Two  Sides  b  and  c,  and 
the  Angle  a  between  them.  — Let  AB  =  c,  AC  =  b^  the  spherical 
angle  BA  C  =  «,  and  the  plane  angle 
BAC  =  «',  0  being  the  center  of  the 
sphere.  About  ^  as  a  center  de- 
scribe a  sphere,  and  let  its  intersec- 
tions with  the  planes  GAB,  OAC, 
and  BAG  form  the  triangle  BUF. 
Then 

I)F=  GAB  =90° 

FDE  =  «  ;  FF=  BAC=a'.  ^^^  ,31 

.  •.  cosFF=  cos  BF  cos  BF  +  sin  BF  sin  BF cos  FBF. 
.-.  cos  «' =  sin -|- 6  sin  I"  c? -f- cos  I  5  cos -|- e  cos  «.  (1) 

This  formula  is  true  for  all  values  of  5,  c,  and  «.  When 
b  and  e  are  small,  the  correction  that  must  be  applied  to  a  to 
obtain  a'  may  be  found  from  (1)  as  follows  : 

Let  p  =  b  -\-  c^  and  q  =  b  —  c.     Then,  from  Art.  72, 
cos  a'  =  l  cos  J  q  —  ^  cos  -|-|)  +  -|-  (cos  \p  +  cos  J  q)  cos  a 

=  —  sin2  \q  -{-  sin2  |-p  +  (1  —  sin^  -^  j;  —  sin^  ^  ^)  cos  a 
=  (sin^  I  j;>  —  sin^  -^  (^)  (sin^  -i-  «  -f  cos^  1  a)  -|-  cos  a 

—  (Qi\i?\p  +  sin2  ^  g)  (cos2  -|- «  —  sin^  |  «) . 
.  ♦.   cos  a'  =  cos  a  —  2  sin^  J  e^'  cos^  1  a  4-  2  sin^  ^jo  sin^  |-  a.  (2) 

*Eq.  (2),  Art.  12L 


l" 


180 


SPHERICAL  TRIGONOMETRY. 


Let  «'  =  «-{-  6^  where  6  is  so  small  that  we  may  place 

sin  6  =  6^  and  cos  6  =  1. 
. '.'  cos  a'  =  cos  a  cos  0  —  sin  a  sin  0. 
.'.   cos  a' =  cos  a  —  0  sin  a.  (3) 

Comparing  (2)  and  (3), 

2  6  sin  1^  a  cos  |- «  =  2  sin^  ^  ^  cos^  i  «  —  2  sin^  ^  jt?  sin^  |^ «. 
.  •.   ^  =  sin2  ^  ^  cot  -|-  a  —  sin^  ^p  tan  -J  a. 

' ''   ^"  ^  ^hTl^  ^''''  *  ^  ''''^  2  «  -  ^i^,-  sinH^  tan  ^  a,         (4) 
since  6?  =  6"' sin  1'^  (Art.  81). 


172.  The  Angles  of  Elevation  of  Two  Points,  in  the  Direc- 
tions OA  and  OB,  above  a  Horizontal  Plane,  and  the  Inclined 
Angle  AOB,  were  measured  with  a  Sextant.  Find  the  Hori- 
zontal Angle  between  the  Points,  as  seen 
from  O.  —  Let  OZ  be  the  vertical  line,  Oah 
the  horizontal  plane;  aOA  =  h^  and  hOB=k 
the  measured  altitudes;  and  AOB=:c  the 
inclined  angle.  Describe  a  sphere  about 
0  as  a  center.     Then  in  the  triangle  AZB, 

AZ  =  90°  -h,  BZ=  90°  -  7c,  AB  =  c, 

and  hence  the  required  angle  aOb  =  AZB 
may  be  computed,  since  we  know  the  three 
sides  of  the  triangle. 

When  h  and  k  are  small,  the  correction 
to  be  applied  to  the  measured  value  c  to  obtain  a  Ob  may  be 
found  as  follows  :* 
From  (2),  Art.  121, 


Fig.  132. 


cos^Z^= 


c  —  sin  h  sin  k 


cose 


hJc 


cos  h  cos  k 
cos  c  —  hk 


(Art.  78) 


1  -  K^'  +  ^') 

.  cos  AZB=  cos  c  +  1(^2  4-  F)  cos  c 


(l-J-A2)(l_l/,2) 

=  (cos  c  -  hk)  [1  +  i  Qi^  +  A;2)] . 

A^.  (1) 


*  Neglecting  powers  of  h  and  k  above  the  second. 


APPLICATIONS  OF   SPHERICAL   TRIGONOMETRY.       181 

Let  6  be  the  correction  to  c  so  that  AZB  =  c  +  ^. 

.  *.  cos  AZB  =  cos  c  cos  6  —  sin  c  sin  6. 

.  •.  cos  AZB  =  cos  c  —  ^  sin  c.  (2) 

Comparing  (1)  and  (2), 

(/,2  ^  ^2)  (cos^  -|-  ff  -  sin2  lc)-2hk  (cos^  ^  g  +  sin^  i  g) 
~~  4  sin  ^  g  cos  ^  g 

.-.   ^=  i(^  +  A:)2tanJg-|(A-/c)2cotl(7,  (3) 

where  ^,  A,  and  A:  are  expressed  in  circular  measure.     To  find 
e  in  seconds,  let  6  =  6"  sin  1",  ^  =  h"  sin  l'^  /c  =  k"  sin  1^'. 

...  B'f ^\Qi''  +y'y&mV'  t2in^c-\(h'' -h''y^mV'  Qot I  c.  (4) 

SPHERICAL  EXCESS. 

173.  Area  of  a  Spherical  Triangle.  —  From  geometry  we 
know  that  the  areas  of  any  two  triangles  are  to  each  other  as 
their  spherical  excesses,  the  spherical  excess  being  the  amount 
by  which- the  sum  of  the  three  angles  exceeds  180°.  We  also 
knoAV  that  the  area  of  the  trirectangular  triangle  is  -|-  Trr^,  and 
that  its  spherical  excess  is  90°.  If  A  is  the  area  of  any  triangle, 
and  E  its  spherical  excess  expressed  in  degrees,  we  have 

Ai\'nr'^=Ei^{i\  (1) 

.-.  A  =  E^^,  (2) 

180°  ^  r 

and  E  =  a'^'  '     (3) 

174.  Lhuillier's  Theorem.  —  We  have  ^V 

tan  1  xT_s^^i(«  +  ^  +  7-'^)    2cos^(ot  +  ^  +  7r-7) 
^      -  ^(jg  1  (^^  +  ^  +  ^  _  ^)    2  cos|(a  +  /S  4-  TT  -  7) 

_  sin  -|-  («  +  /3)  —  sin  j-  (tt  —  7) 
""  cos  I  (a  +  yS)  +  cos  ^  (tt  ~  7)' 

from  (6)  and  (7),  Art.  72. 

^  cos  i  («  +  y8)  4-  sm  J  7 


182  SPHERICAL  TRIGOXOMETRY. 

Hence,  from  (1)  and  (2),  Art.  145,  substituting  for  sin^(a-f-/3) 
and  cos  ^  (a  4-  yS),  we  have 

-  ^     cos  i  (a  —  h)—  cos  i  c     cos  X  y 

tan  i  J5;= \^ — —j^ f ^-f-i- 

*         cos  ^  (a  +  6)  +  cos  J  <?     sin  J  7 

^  sinK^'^-^  +  ^)si"i(^  +  g-a)  ^^^  1 
cos  :^  (a  +  6  +  0  cos  ^  (a  +  6  —  c)        2"  v» 

from  (4)  and  (3),  Art.  73. 


,  _      sin  J  (s— 5)sinJCs  —  «)^  /     sin  s  sin  (s  —  f?) 

.-.  tani^= ^ "^-T-/— — ^\-^— 7 ^       r 

*  cos  ^  s  COS  ^Qs  —  c)      ^  sm  («  —  a)  sin  (s  — 


4 


sin2  j-  (g  -  ^>)  sin^  1  (g  -  a)  ^^ 

COS^  -1-  8  COS^  J  (S  —  t?) 


sin  J  s  cos  J  s  sin  J  (s  —  c)  cos  ^  (s  — 
sin  ^  (s  —  a)  cos  J  (s  —  «)  sin  |^  (s  —  ^)  cos 


f) 1. 


.   tan  ^  -E'=  Vtan  |^  s  tan  |^  (s  —  a)  tan  |^  (s  —  ^)  tan  ^  (s  —  c?).  Q.E.i. 

175.  Spherical   Excess   in   Terms   of  Two   Sides   and  their 
Included  Angle. 

tan  ^  J?  =r  ^""^  2  («  +  /^  +  7  -  tt)  ^  -  cos-|-  («  +  ^  +  7) 
2  cos  1  (a  4-  ^  +  7  —  tt)         sin  |-  (a  +  /3  +  7) 

_  sin  I  (a  +  /3)  sin  j^  7  —  cos  j-  (a  4-  /5)  cos  j-  7 
""  siji  ^  (a  +  13}  cos-|-  7  +  cos  ^  («  +  fi)  sin  J  7 

Substituting  for  sin  |-  («  +  /3)  and  cos  J  (a  +  /3)  from  (1)  and 
(2),  Art.  145, 

^  ^  _  sin  ^  7  cos  I  7[cos  J  (a  —  h)  —  cos  |^  («  +  ^)] 
2       "~  cos|^(rt  —  6)  cos2^7  4- cos  J(a  +  6)  sin2l7 

sin  -|-  7  cos  -|-  7 f  4-  2  sin  J  a  sin  ^  />! 
~^[cos^(a  — 6)4-cos^(a4-^)J  +  -2-[cos^(^a— 6)  — cos-^(«4-^)]cos7 

_  sin  I"  a  sin  J  5  sin  7 

cos  ^  a  cos  ^b  -{-  sin  ^  a  sin  J  6  cos  7 

...  tanA^=,  tan^atan^ismy    .  ^.^.l. 

"*  1  4-  tan  J  a  tan  J  6  cos  7 

176.  Approximate  Value  of  the  Spherical  Excess,  neglecting 
Powers  above  the  Second.  —  Let  the  sides  of  the  triangle  be  so 


APPLICATIONS  OF   SPHERICAL  TRIGONOMETRY.       183 

small  that  the  powers  of  their  circular  measures  higher  than 
the  second  may  be  neglected.     We  have,  from  Art.  78, 

tana;  =  a; -I- Ja;3  4- •••,  (1) 

where  x  is  expressed  in  circular  measure. 

Let  the  lengths  of  the  sides  be  a,  6,  and  c  when  expressed 
in  circular  measure,  and  a\  h\  and  c'  in  linear  measure,  r  being 
the  radius  of  the  sphere.     Then 

«  =  p     ^  =  p     ^=7  (2) 

Placing  these  values  of  a,  6,  and  c  for  x  in  (1),  and  substituting 
in  Lhuillier's  theorem,  we  have,  neglecting  powers  above  the 
second, 

for,  1  7?     ^V^^      s'  —  a'     &'  —  V     s'  —  c'  ,oN 

where  s' =  ^(a' +  5' +  c').  (4) 


.-.  tani^  =  -!-Vs'(s'  -  a')(«'  -  ^-OC^'  -  O-  C^) 

4  H 

Since  \Il\s,  small,  we  place  its  tangent  equal  to  its  arc. 

.-.  \E=^  A^«'(«'  -  ^')(s'  -  ^')0'  -  O  (6) 

4H 

^  =  1^,  (7) 

where  ^  is  the  area  of  the  plane  triangle  whose  sides  are  a',  V ^ 
and  c\  E  being  expressed  in  circular  measure. 

To  find  the  value  of  E  in  seconds  of  arc,  divide  both  sides 
by  sin  1''. 

=  En  =  —A_.  (8) 


E 


sin  1  f  f  r^  sin  1 " 

Hence,  whenever  the  third  powers  of  the  circular  measures 
of  the  sides  can  be  neglected,  the  spherical  excess  is  found  by 
computing  the  area  of  the  triangle,  considering  it  as  a  plane 
triangle,  and  dividing  the  area  by  r^miV . 


184  SPHERICAL   TRIGONOMETRY. 

177.    Approximate  Value  of  the  Spherical  Excess,  neglecting 
Powers  above  the  Fourth.  —  From  Lliuillier's  theorem, 

*  L2r      24?-aJL    2r     ^       24r3     J 

fs'  -  b'       (s'  -  hy~\[s'  -  c'      (js'  -  gpn 
L    2r  24^3     JL    2r  24?-3     J 


where  A^  =  s'(.s'  -  a')(^'  -  ^00'  -  ^')- 

.4^  A^ 

16  r*      192  r" 


4r2V               24?-2        / 
•••  i^'s^^^l'   =4^(,^-^ 247^ ^j- 

r2sinl"V  24  r2        J  ^ 

This  value  exceeds  that  found  in  Art.  176  by 
A         a^2  ^  yi  4.  ^/2 
r2sinl''*  24  r2 

If  a'  =  b'  =  c'  =  100  miles,  and  r  =  3963.3  miles,  we  obtain 

-^^— -  =  56^^863;     ^!-^tJ!l±^  ^qmOOS; 
?-2  Sin  1'^  24  H 

so  that  the  correction  to  the  value  of  U''  given  by  (8),  Art. 
176,  is  only 

56'^863  X  0.00008  =  0'^003. 

178.  Legendre*s  Theorem.  —  If  the  sides  of  a  spherical  tri- 
angle are  veri/  small  compared  ivith  the  radius  of  the  sphere,  the 
angles  of  the  plane  triangle  whose  sides  are  of  the  same  length  as 


APPLICATIONS  OF  SPHERICAL  TRIGONOMETRY.       185 

those  of  the  spherical  triangle,  are  equal  to  the  corresponding 
angles  of  the  spherical  triangle  diminished  hy  one  third  of  the 
spherical  excels.  —  Let  a',  h\  and  c'  be  the  lengths  of  the  sides 
of  the  spherical  triangle  expressed  in  linear  measure,  and  a,  6, 
and  c  the  lengths  in  circular  measure. 

a'       J       h'  c'  .^^ 

r  r  r 

Let  a  be  an  angle  of  the  spherical  triangle  and  a'  the  corre- 
sponding angle  of  the  plane  triangle.     We  have 

cos  a  —  cos  b  cos  c 


cos  a  = 


(2) 


cosa  =  -V[(^2  +  c2_^2) 
2  be 


sin  b  sin  c 
From  Art.  78, 

cos  a  =  1  —  I  a2  -f  2^  a*  —  •••  sinb  =  b  —  -J-  b^  +  ... 

cos6  =  1  -  -J62  +  ^\b^ smc  =  c  -  i6'3  +  ... 

cos  (?  =  1  —  -J  c2  +  2f  <?^  —  ••• 

X  (1,2  _!_  ^2  _  ^2)  +     1    (^4  _  54  _  .4  _  e  J2^2) 

•*•   cosa  =  ^^ j-jr — ^t-770  ,     0.-1 ^'    (^) 

the  terms  of  orders  higher  than  the  fourth  being  neglected. 

[(52  +  ^2  _  ^2) 

+  J^  (a4  _  J4  _  ^4  _  e  52^2)-|  [-1  _  ^  (^2  +  ^2)-]-l 

+  _i_ (a4  _ /,4  _  ^4  _  6  62^2)]  [1  4- K^' +  ^')  +  •••] • 

52  4.  ^2  _  ^2       ^4  +  54  _f_  ^4  _  2a2^2  _  2a\^  -  llP-c^ 
.-.    cos  «  =        \   +  — ^- ^ -— , 

2bc  24:  be  /'4\ 

the  terms  of  orders  higher  than  the  fourth  being  neglected,  as 
before. 

In  the  plane  triangle, 

^         ^'2  +  ^r2_^/2        J2  4.^,2_^2 

^^^^=         2  6V         =         2bc       '  ^^^ 

from  (1) 

,         «4  +  ^4  +  ^4  _2a2^»2_  2^2-2^2^2 

.-.  cosa  =  cosa'H ■ — .  (6) 

24  oc 

,      1     a'4  +  5'4  4-c'4_2a'26'2_2a'2e'2-2  6V2 
...   cos«  =  cos«+- ^j^,-^ 


1S6  SPHERICAL   TRIGONOMETRY. 

Let  8'  =  Ka'  +  b'  -\-  c')  ;   then 

=  -  iV  («'^  +  ^"  +  ^'^  -  2  a'25^2  _  2  a'V2  -  2  5'V2).     (8) 
But  the  area  of  the  plane  triangle  is 


Vs'(s'-a')(s'-6')(s'-60  =  ^  6'tf'  sin  a^  (9) 

Hence  (7)  becomes,  from  (8)  and  (9), 

cos  a  =  cos  a'  —  r—7,h'c'  sin^a'.  nO") 

6  r^  ^ 

Let  a  =  a'  +  ^.  (11) 

.-.  cos  a  =  cos  a' cos  ^  —  sin  a' sin  ^.  (12) 

Since  6  is  small,  we  may  place  cos  ^  =  1,  and  sin  6  =  d. 

.  *.   cos  a  =  cos  a'  —  0  sin  a^  (13) 

Comparing  (10)  and  (13), 

a        ^    ti  I    •      /      11     b'c' sin  a'  ^^.^ 

e  =  — 6Vsm«'=-.-._^-.  (14) 

Hence,  from  (7),  Art.  176, 

and,  from  (11),  a'  =  a  —  ^ K  Q.B.D. 

179.  Application  of  Legendre*s  Theorem.  —  In  the  New  York 
State  Survey  the  angles  of  the  spherical  triangle,  whose  ver- 
tices were  at  Howlett,  Gilbertsville,  and  Eagle,  were  measured, 
the  distance  from  Howlett  to  Gilbertsville  having  been  already 
computed.     The  measured  values  were 

At  Howlett,  a  =    85°  18'  57". 71         logh  =  4.54227  32 

At  Eagle,  /3  =    51°  35' 41''.61         logr=  6.80459  32 

At  Gilbertsville,         7=    43°    5' 24". 24 

...  « -1-/3  +  7  =  180°    0'    3". 56 
The  formula  for  the  spherical  excess  is  (Art.  176) 
■pn  _        ^        _  1     52  sin  a  sin  7     1 


r^sinl"      2  sinyS  r'^sinV 


APPLICATIONS  OF   SPHERICAL   TRIGONOMETRY.      187 

log52  =  9.08455 
colog  2  =  9.69897 -10 
logsina  =  9.99855 -10 
log  sin7  =  9.83451  -10 
colsin/3  =  0.10588 
cologr2=  6.39081 -20 
colsinr'  =  5.31443 

logU'f  =  0.42770  .-.  E"  =  2''. 6773; 

.-.  1^^' =  0'^8924. 

The  errors  due  to  observation  therefore  amounted  to  3''. 56 
—  2''. 677  =  0''.883.  This  discrepancy  was  distributed  among 
the  three  angles  according  to  the  method  of  least  squares,* 
gfving  the  following  results  : 


Observed 

Angles. 

Correction. 

Spherical 
Angles. 

i^". 

Plane  Angles. 

o=   85°  18' 

57".71 

-0''.747 

56".963 

0".893 

56".070  =  a' 

jS  =    51°  35' 

41".61 

+  1".355 

42".965 

0".892 

42".073  =  ^' 

7=    43°   5' 

24".24 

-  1".491 

22".  749 

0".892 

2l".8J7  =  7' 

Suni=180°   0'  3".56         -0".883         2".677  2".677  0".000 

Using  the  plane  triangle,  we  find  by  the  sine  proportion  : 


log  5 
col  sin  /S 
log  sin  a 

log  a 
a 


=  4.542  2732 
=  0.105  8837 
=  9.998  5468 


10 


4.646  7037 
44330.61  meters. 


log5'  =  4.542  2732 
colsinyQ'  =  0.105  8837 
log  sin  y  =  9.834  5089-10 

log(?'  =  4.482  6658 

c'  =  30385.46  meters. 


These  are  the  distances  between  the  points  measured  on  the 
great  circles  joining  them. 

ASTRONOMICAL  APPLICATIONS. 

180.  Definitions.  —  Let  us  consider  the  earth  as  a  point  0 
(Fig.  134),  and  let  a  sphere  be  described  about  0  as  a  center, 
with  a  radius  indefinitely  great,  so  that  all  the  stars  shall  be 
within  the  sphere.  The  figure  represents  the  sphere  as  seen 
from  the  outside. 


Eleven  angles  were  involved  in  the  adjustment. 


188 


SPHERICAL   TRIGONOMETRY. 


The  zenith  Z  is  the  point  where  a  vertical  line  —  the  plumb 
line  —  pierces  the  sphere. 

The  horizon  HWNE  is  the  great  circle  cut  from  the  sphere  by 
a  plane  through  0  perpendicular  to  the  plumb  line.  iV,  E,  H^ 
and  TF^are  the  north,  east,  south,  and  west  points  of  the  horizon. 


^"7-   '  ^ 


FiQ.  134. 


Vertical  circles  are  great  circles  whose  planes  pass  through 
the  plumb  line  OZ,  as  ZST  in  the  plane  OZT. 

The  meridian  HZN  is  the  vertical  circle  passing  through 
the  north  and  south  points  of  the  horizon. 

The  altitude  TS  of  a  star  or  point  is  its  angular  distance 
above  the  horizon,  measured  on  a  vertical  circle. 

The  zenith  distance  ZS  is  the  complement  of  the  altitude. 

The  azimuth  of  a  star  or  point  is  the  arc  NT  or  the  angle 
NZT  between  the  meridian  and  the  vertical  circle  through  the 
star  *  or  point.  It  is  usually  measured  from  the  south  point  of 
the  horizon  through  the  west. 

The  poles  P  and  P'  are  the  intersections  of  the  axis  of  the 
earth  with  the  sphere.  P  is  here  the  north  pole.  In  conse- 
quence of  the  earth's  rotation  about  its  axis  the  stars  appear  to 

*  That  is,  whose  plane  passes  through  the  star. 


APPLICATIONS  OF   SPHERICAL   TRIGONOMETRY.       189 

describe  small  circles  about  P  as  the  pole,  apparently  moving 
in  the  clirection  EQWQ', 

The  equator  EQ  WQ'  is  the  great  circle  cut  from  the  sphere 
by  a  plane  through  0  perpendicular  to  the  axis  of  the  earth. 

The  latitude  of  the  observer  is  the  angular  distance  QZ  from 
the  equator  to  the  zenith.  Since  PQ  =  90°  and  ZN  =  90°,  we 
liave  NP  =  QZ^  i.e.  the  elevation  of  the  pole  above  the  horizon 
is  equal  to  the  latitude  of  the  place. 

The  hour  circle  of  a  star  is  the  great  circle  PSD  through  the 
star  *  and  the  pole.  All  the  hour  circles  are  perpendicular  to 
the  equator. 

The  hour  angle  of  a  star  is  the  angle  at  the  pole  between  the 
meridian  and  the  hour  circle  of  the  star,  measured  from  the 
meridian  to  the  west.  Thus  the  hour  angle  of  S  is  —  ZPS^ 
negative  since  it  is  measured  to  the  east.  It  is  so  named 
because,  if  the  angle  ZPS  is  15°,  one  hour  will  elapse  before 
PS  coincides  with  PZ ;  for  15°  =  360°  --  24,  and  the  star 
appears  to  make  a  complete  revolution  about  P  in  24  hours  of 
sidereal  (^i.e.  star)  time. 

The  declination  DS  of  a  star  is  its  angular  distance  from  the 
equator,  measured  on  its  hour  circle,  and  positive  when  the  star 
is  north  of  the  equator. 

The  right  ascension  of  a  star  is  the  angular  distance  along 
the  equator  from  a  certain  point  on  the  equator,  called  the  ver- 
nal equinox^  to  the  foot  of  the  hour  circle  through  the  star, 
measured  towards  the  east ;  or  it  is  the  angle  at  the  pole  be- 
tween the  hour  circle  of  the  vernal  equinox  and  that  of  the 
star. 

Hence  the  angle  between  the  hour  circles  of  two  stars  is 
equal  to  the  difference  between  their  right  ascensions. 

181.  At  a  Place  in  Latitude  42°  N.  the  Altitude  of  a  Star, 
whose  Declination  is  +  60°,  was  measured  and  found  to  be 
50°,  the  Star  being  East  of  the  Meridian.  At  what  Time  did 
the  Star  reach  the  Meridian?  —  In  the  triangle  ZPS,  ZP  =  48°, 
ZS  =  40°,  PS  =  30°  ;  .  •.  by  Art.  148,  -|-  ZPS  =  29°  55^9  ; 
.-.  ^^.9=  59°  51'. 8  or  3*  59^5.  Hence  the  star  reached  the 
meridian  3*59™  .5  after  the  observation  was  made. f 

•  That  is,  whose  plane  passes  through  the  star.        t  Sidereal  time. 


1^0 


SPHERICAL   TRIGONOMETRr. 


182.  The  Latitude  of  the  Place  being  42°  N.,  find  the  Interval 
of  Time  between  the  Rising  of  a  Star  above  the  Horizon  and  its 
Passage  across  the  Meridian,  its  Declination  being  +  10°.  —  In 
the  triangle  ZPS,  S  will  be  on  the  horizon  NEH  at  the  instant 
of  rising,  so  that  ZS  =90°.      - 

.  •.  cos  ZaS'  =  0  =  cos  ZP  cos  SP  +  sin  ZP  sin  SP  cos  ZPS. 

.  •.  cos  ZPS  =  -  cot  ZP  cot  SP  ^-  cot  48°  cot  80°. 

.-.  ZPS=m°  8'.ror  6*  36'".5.* 

Hence  the  star  will  be  abcute_the  horizon  13*  13"*.0.* 

183.  The  Latitude  of  the  Place  being  42°  N.,  and  the  Declina- 
tion of  the  Star  +  20°,  find  the  Interval  between  the  Instant 
when  it  is  due  East  and  that  when  it  is  due  West.  —  In  the 
triangle  ZPS,  PZS  =90°. 

.  •.  cos  ZPS  =  tan  ZP  cot  SP  =  tan  48°  cot  70°. 

.-.  2^P^=6G°  9^4.    .-.  2Zi^AS'=132°  18'.8  =  8*49'».3. 

Hence  the  interval  required  is  8*  49'".3.* 

184.  The  Latitude  being  42°  N.  and  the  Declination  of  the 
Star  +  80°,  find  the  Azimuth  of  the  Star  when  it  is  at  its 
Greatest  Western  Elongation ;  that  is,  when  the  Star  has  reached 

its  Farthest  Distance  towards 
the  West,  afterwards  moving 
East.  —  In  the  figure  the 
ZPS  triangle  is  projected 
upon  the  plane  of  the  hori- 
zon, so  that  Z  is  the  zenith, 
P  the  pole,  S  the  star, 
MSM'  the  apparent  diurnal 
path  of  the  star  about  the 
pole,  ZP  the  meridian,  ZS 

the  vertical  circle  of  the  star,  and  PZS  the  angle  required,  the 

angle  ZSP  being  a  right  angle. 


Fig.  135. 


*  Sidereal  lime. 


APPLICATIONS  OP  SPHERICAL  TRIGONOMETRY.      191 

.  •.  sin  SF  =  sin  ZP  sin  PZS.     .  •.  sin  FZS  =  sin  10°  cosec  48° ; 

.-.  PZaS'=13°30'.8. 

Note.  —  This  is  the  method  ordinarily  used  by  the  engineer  to  determine 
the  north  and  south  line. 

185.  The  Right  Ascensions  of  Two  Stars 
are  a  and  a',  and  their  Declinations  6  and  8' ; 
find  the  Angular  Distance  between  the  Two 
Stars. 

SP=90°-B,   S'P=:90°-S',   SPS'  =  a'-a. 

Hence  we   know  two   sides  and  the   in- 
cluded angle,  and  we  find  the  third  side  SS'   ^        fig.  m. 
by  Art.  151  or  by  Art.  150. 

186.  If  a'  and  a'^  are  the  Right  Ascensions,  and  h'  and  8''  the 
Declinations  of  Two  Stars,  find  the  Inclination  to  the  Equator  of 
the  Great  Circle  passing  through  the 
Stars,  and  also  the  Right  Ascension  of  ^J^ 
the  Point  where  it  cuts  the  Equator.  —  ^^ 
Let  B  and  D  be  the  two  stars,  UQ  v^e  'a 
the   equator,   V  the  vernal  equinox,                   fiq.  ist. 

E  the  intersection  of  the  great  circle  BD  with  the  equator, 
VE  =  ay     In  the  right  triangle  EAB, 

sinEA  =  tiinAB  cot  AEB.      .-.  cot^=sin(a^  — «i)cot  S'.     (1) 

In  the  right  triangle  ECD^ 

sin  EO=tiiii  CI)  cot  CED.      .-.  cot  z  =  sin(«''-«i)cot  8'^    (2) 

sin  («"  —  wj)  _  cot  8' 
sin  ((x'  —  cii)  ~  cot  h" 

.  ^    sin  (f/^  -  «^)  +  sin  {a'  -  a^  _  cot  h'  +  cot  h^' 
sin  («"  —  «i)  —  sin  («' —  «j)  ~~  cot  8' —  cot  3'' 

,.    tan  I  (ja'  ^  +  r/  -  2  a^^  _  sin  (h"  +  8^) 
tan !-(«"-«')       ~sin(8''-S') 

...  tanK«'^  +  «'-2«0=|^§^^tanlC«''-«0.      (3) 

From  (3)  find  \(^a'^  +  «'  —  2  Wj),  thence  finding  a^i   then  i 
may  be  found  from  either  (1)  or  (2). 


192 


SPHERICAL   TRIGONOMETRY. 


187.  The  Right  Ascension  and  Declination  of  a  Star  are  a  and 
8,  and  those  of  Another  Star  are  a'  and  5' ;  find  the  Hour  Angle 
Ox  the  First  Star  and  their  Common  Azimuth  when  the  Stars  are 
^g  in  the  Same  Vertical  Circle,  the  Lati- 
tude of  the  Place  being  <().  —  There 
are  two  positions,  one  when  both 
stars  are  west,  and  the  other  when 
they  are  both  east,  of  the  meridian. 

(1)  S'F  =  dO°-S';  SF  =  90°-^; 
SPS'  =  a-a'\  ZP  =  ^T-(i>.  In  the 
triangle  SPS\  find  P^S^^S'  and  PSS'. 
Then  in  the  triangle  S'PZ  we  know 
S^P,  ZP,  and  PS'Z,  and  Ave  find 
PZS'  and  ZPS'.  In  the  triangle 
SPZ  we  know  SP,  ZP,  and  PSZ  = 
180°  -  PSS',  and  Ave  find  PZS  and  ZPS, 

The  checks  are  PZS'  =  PZS,  and  S' PZ-SPZ=:a-«! . 
(2)  S^P  =  90°  -  S,  S^P  =  90°  -  a',  S^PS^  =  ar-a';  find 
PS^S^'  and  PS^'S-^^,  these  angles  being  the  same  as  those  at 
S  and  S'  in  the  first  case.  Then  from  the  tAvo  triangles  PS^Z 
and  PS^'Z  Ave  find  the  angles  PZS^  and  PZS^',  Avhich  should 
bs  identical,  and  also  the  angles  S^PZ  and  S-^'PZ,  Avhose  dif- 
ference should  be  a  —  a'. 


"-•^^l     Ir^ 


LOGARITHMItr  AND  iRIGONOiMETRIC 


TABLES 


FIVE   DECIMAL    PLACES 


EDITED    BY 


C.    W.    CROCKETT 

PROFESSOR   OF  MATHEMATICS  AND  ASTRONOMY 
RENSSELAER   POLYTECHNIC   INSTITUTE 


J'i9^<^ 


NEW  YORK  •:•  CINCINNATI  .:•  CHICAGO 

AMERICAN    BOOK    COM^^amv 


:92 


CONTENTS. 


PAGH 

3 


Table     I.   Logarithms  of  Numbers 

S\T',S'\T",ioT2r-s'      ......  24 

II.    Logarithms  of  Trigonometric  Functions      .     .  25 

III.  Natural  Trigonometric  Functions      ....  71 

IV.  Lengths  of  Circular  Arcs 9- 

V.    Conversion  of  Logarithms 0 

Formulas q- 

Constants j^- 

Explanation  of  the  Tables 10- 


NoTE.  —  The  well-known  tables  of  Gauss,  Becker,  and 
/■il^.Tr^t  have  been  taken  as  the  standards,  the  proof  sheets 
have  been  read  with  gTCat  care,  and  it  is  believed  that  the 
numl)er  of  errors  cannot  be  iar°^e.  The  arrangement  of 
the  figures  on  the  page  is  in  accordance  v'ith  that  adopted 
in  the  standard  six  and  seven  place  tables. 

'I'he  natural  tables  were  reduced  from  seven-place  tables 
and  compared  with  published  five-place  tables. 

For  convenience  in  using  the  tables,  the  explanation 
'  ten  placed  after  them  instead  of  before  them. 


\KIGHT,   1896,   BY   AMERIC-VN    3oOK 


■ 

3 

\ 

f'^         >\ 

( 

.    UNIVERSITY  ] 

I. 

COMMON 

LOGARITHMS    OF 

NUMBERS 

' 

•  ' 

FROM     I     TO     I  1000. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

N.          Log. 

0 

2 

3  * 

— 

20 

21 
22 
23 

1.30  103 

1.32  222 
1.34242 
1-36173 

40 

41 
42 

43 

1.60206 

60 

61 
62 
63 

1-77815 

80 

1.90309 

O.OO  CXXD 
0.30  103 
0.47712 

1.61  278 

1.62  325 
1-63347 

1-78533 
1.79  239 

1-79  934 

81 
82 

1.90849 
1.91  381 
1.91  908 

4-, 

0.77815 

24 

25 
26 

1. 38  021 

1-39  794 
1.41  497 

44 
45 
46 

.1-64345 
1.65321 
1.66276 

64 

65 
66 

1.80  618 

1.81  291 
1.81  954 

!4 

8^^ 

1.92428 
1.92942 
1.93450 

7 
8 

9 
10 

i  t 

0.84510 
0.90309 
0,95  424 

1. 00  000 

r.04 139 
1.07  918 
1.11 394 

27 

28 
29 

30 

31 

11 

1-43  136 
1.44716 
1.46  240 

49 
50 

51 

52 
53 

1.67  210 

1.68  124 
1.69020 

67 
68 
69 

70 

71 

72 

73 

1.82607 
1-83251 

1.83885 

1.84  510 

89 

90 

91 
92 

93 

1-93952 
,1-54448 
1-94539 

M77I2 
1.49- 

1.50  i  . 
1.51851 

1.69897 

1.95  424 

1.70757 
1. 7 1  600 

1.72428 

1.85  126 

1.85  733 
1.86332 

1.95904 
1.96379 
1.96848 

H 

IS 

i6 

1.14613 

1.17609 
1.20412 

34 

11 

1-53  148 
1.54407 

1-55630 

54 

11 

1-73239 
1.74036 
1.74  819 

74 
75 
76 

1.86923 
1.87506 
1.88081 

94 
96 

1-97313 

1.97772 
1.98227 

17 
i8 

1  20 

1.23045 

1-25  527 
1.27875 

1.30 103 

37 
38 
39 

40 

1.56820 

»-57  978 
1.59  106 

57 
58 
59 

60 

1-75  587 
1-76343 
1.77085 

77 
78 
79 

80 

1.88649 
1.89  209 
1.89763 

1.90309 

97 
98 
99 

100 

1.98677 
1.99  123 
1.99564 

1.60  206 

1.77  815 

2.00000 

o 

I 

S'. 

'            6.46  371 

373 

T. 

37:^ 
37  ^ 

0°    0'  - . 
0      I     - 
0      i   —  1 

0" 
60 
20 

S".               T". 

4-68  557              557 

557             557 

.^57             557 

S'.  T'.J    N. 

L.    0 

1         2 

3 

4 

5 

6 

7 

8    '    9 

P.  P.  ^ 

6  "'■'     f 

..  ,^^_ 

366' 

3.= 

iDO 

00000 

043 

087 

130 

^73 

217 

260 

303 

346 

389 

44    43     42 

366 

38s  j   'Ol 

432 

475 

578 

561 

604 

/647 
♦072 

689 

732 

775 

817 

366 

38,      i02 

860 

903 

945 

988 

♦030 

*ii5|*i57 

*i99 

*242 

J     ^  .     .  ^    ,.  „ 

^•4     4- j     4--^  ' 

8.8    8.6  8.4 
13.2  12.9  12.6 
17.6  17.216.8 

366 

•  366 

3&''     1 03 

01284 
703 

326 

745 

3^8 

787 

410 

828 

452 
870 

494 
912 

536 
^11 

578 
995 

620 
♦036 

662 

*078 

2 

3 

4 

366 

X^i-     \  05 

02  119 

I  bo 

202 

243 

284 

325. 

366 

407 

449 

490 

5 
6 

22.0  21.5  2 f.o 
26.425.825.2 

366 

3«n 

io6 

531 

572 

612 

653 

694 

735 

776 

816 

857 

898 

366 

3S7 

',07 

938 

979 

♦019 

*o6o 

*IOO 

*i4i 

*i8i 

♦222 

*262 

*302 

7 

30.8  30.1  2C.4 

365 

387 

108 

03342 

383 

423 

463 

503 

543 

583, 

623 

663 

703 

8 

35.2  34.4  33.6 

365 
'365 

I365 

387 
^7 
3S8 

T09 
III 

743 

J^2^ 

822 

862 

902 

941 

9^1 

*02I 

*o6o 

*IOO 

9 

39-6  38.7  37-8 

04i39_ 

J79_ 

218 

258 

"297^ 

336 

~37^ 

415 

454 

493 

41     40    39 

532 

571 

610 

650 

689 

727 

766 

805 

844 

883 

365 

388 

112 

922 

961 

999 

*038 

*o77 

*ii5 

*^54 

*192 

♦231 

*269 

^ 

4.1  4.0    3.0 

8.2  8x3    7.:,, 

365 

38S 

"3 

05  308 

346 

385 

423 

461 

500 

538 

576 

614 

652 

2 

■' 

36S 

389 

114 

690 

729 

767 

805 

843 

881 

918 

956 

994 

♦032 

3 
4 

12.3  12.0  11.'^ 

16.4  16.0  15.61 

20.5  20.0  19.5 

24.6  ?  1  n  23,4  1 

365 

389 

115 

06070 

108 

145 

185 

221 

258 

296 

333 

371 

408 

364 

389 

116 

446 

483 

521 

558 

595 

633 

670 

707 

744 

781 

5 
6 

364 

389 

ri7 

819 

856 

893 

930 

967 

*oo4 

♦041 

*o78 

*ii5 

*i5i 

7 

28.7  28.0  27.3 

364 

390 

118 

07  188 

225 

262 

298 

335 

372 

408 

445 

482 

518 

8 

32.832.031.2 

364 

-  364 

390 

'I9 
120 

555_ 
918 

59t 
954 

628 

6b4 

700 

737 
*099 

773 
*i35 

809 

846 

882 

9 

36.9  36.0  3^.1 

990 

♦027 

*o63 

*i7i 

*207 

*243 

38    37    36 

364 

391 

121 

08  279 

3H 

350 

386 

422 

458 

493 

529 

565 

600 

363 

39' 

J  22 

636 

672 

707 

743 

778 

8.4 

849 

884 

920 

955 

I 

3.»    3-7    3-^\ 

363 

39  ■ 

123 

991 

*026 

*o6i 

♦096 

*I32 

*i67 

♦202 

*237 

*272 

*307 

2 

7.6    7.4    7-2' 

363 

39  r 

124 

09342 

377 

412 

447 

482 

517 

552 

587 

621 

656 

3 

11.4  11. 1  10.^' 
15.2  14.8  14./ 
19X)  18.5  18.0 

22.8  22.2  2I.( 

3<^3 

392 

125 

691 

726 

760 

795 

830 

864 

899 

934 

368 

♦003 

4 
5 
6 

,363 

352 

126 

10037 

072 

106 

140 

ns 

209 

243 

278 

312 

346 

,36, 

392 

127 

380 

415 

449 

483 

517 

551 

585 

619 

653 

687 

7 

26.6  25.9  25.2 

I363 

393 

128 

721 

755 

789 

823 

857 

890 

924 

958 

992 

♦025 

8 

^0.4  29.6  28.8 

362 
362 

393 
393 
394 

129 
I30 

131, 

II  059 

093 

126 

160 

193 

227 

261 

294 

327 

361 

9 

34-2  33-3  32.4' 

394 

727 

428 

461 

494 

528 

561 

594 

628 

661 

694 

35     34     33 

760 

793 

826 

860 

893 

926 

9t;Q 

992 

♦024 

\lt.. 

394 

132 

120^7 

090 

123 

156 

189 

222 

254 

287 

320 

352 

I 

3-5    3-4   3-3 

362 

3;>4 

^^2, 

385. 

4j8 

450 

483 

516 

548 

581 

613 

646 

678 

2 

7.0   6.8    6.6 

362 

- 
395 

134 

710 

743 

775 

808 

840 

872 

905 

937 

969 

*OOI 

3 

10.5  10.2   9,9 
14.0  13.6  13.2 
17.5  17.0  16.5 
21.0  204  19.8 
24.5  23.8  23.1 
28.0  27.2  26.4 
^i  c  -in  6  00-7 

361 

393 

J  35 

13033 

066 

oq8 

130 

162 

194 

226 

258 

290 

322 

4 

361 
361 

395 
396 

136 
U7 

*   354 
672 

386 
704 

418 
735 

450 
767 

481 
799 

513 

830 

545 
862 

577 
893 

609 
925 

640 
956 

5 
6 

361 

396 

138 

988 

*oi9 

*oqi 

♦082 

*ii4 

*I45 

*i76 

*2o8 

*239 

♦270 

361 

396 

139 

14301 

333 

364 

^1 

426 

457 

489 

520 

551 

582 

Q 

361 
360 

397. 
397 

140 

141 

613 
922 

644 
953 

675 
983" 

706 

"737" 
*045 

768 
*076 

799 
*io6 

829 
*'37 

860  ]  891 

32     31     30 

*oi4 

*i68 

*i98 

i;6o 

397 

142 

11;  229 

259 

290 

320 

351 

381 

412 

442 

473 

503 

I 

3-2    3-t    3-0 

\yo 

393 

143 

■^534 

564 

594 

625 

655 

685 

715 

746 

776 

806 

2 

6.4    6.2   6.0 

J360 

398 

144 

836 

866 

897 

927 

957 

987 

*oi7 

*047 

*077 

*io7 

3 

9.6   9.3    9.0 

I360 

398 

^45 

16137 

167 

197 

227 

-2-56- 

286 

316 

346 

376 

406 

4 

12.8  12.4  12.0 

:     360- 

399 

146 

435 

465 

49? 

524 

554 

584 

613 

643 

673 

702 

5 

'^•°  "il  Tn 

359 

399 

147 

732 

761 

791 

820 

850 

879 

909 

938 

967 

997 

6 

19.2  18.6  18.0 

359 

399 

148 

17026 

056 

085 

114 

143 

^73 

202 

231 

260 

289 

7 
8 

22.4  21.7*21.0 
25.6  24,8  24.0 

359 
359 

400 
400 

149 
ISO 

'  3-     348 
609     538 

377 

406 

435 

464 

493 

522 

81  r 

551 

580 

667 

696 

725 

754 

782 

840 

869 

9  ^^'"^  -^i'y  ^/-^ 

N. 

L.    0  1     1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

S.'      T.' 

II                                      S."     T." 

S."    T." 

I'    646  373     373 

0°     i'=     60"  4.68557     557 

0°  19'=  1 140"    4-68557     558,. 

2             373     373 

0      2  =   120             557      557 

0    20=1200              557     558 
0    21  =1260              5f7     50 
0    22=1320              557     5ju 

'  '■'  "^'           :■  /  J 

0        3  =    180                   ^K-j       557 

i73 

0      16  =    960                  557       558 

:  "            N>73 

0      17  =IC20                    557        558 

c    23=1380  ■     •       557     558 

^3     t::\ 

11  0      18  "=:I0S0                     S"^/        558 

0    :r!  =1440               557     558 

11    , 

,r.    — 

f  1  »r» 

c^7 

<;•,? 

0    i 

5  =^ 

500 

557     558  1 

^ 

.(_ 



_,_ 

b  .   T'.' 

N~ 

L.   0     1. 

2    1    3    1   4 

5    1    6 

7 

8 

9 

p.  p.      1 

1     ■■'■ 

400 

40t 

150 

15' 

I7OO9 
898 

9:^13 

667 

OoOi    725 

754 
♦041 

*070 

811 
*«S99~ 

840 

*I27 

869 
*I56 

29       9.^ 

i|  2.9 
2!  5.8 

3i  8.7 

4!ii.6    1           i 

955 

984  1*013 

401 

IS2 

18184 

2K3 

241 

j/O  ,   298 

327 

355 

384 

412 

441 

401 

^53 

469 

498 

526 

55-J  1  583 

611 

<539 

667 

696 

724 

402 

154 

752 

780 

808  1  8:57  !  S65 

893 

921 

,  949 

977 

*cx)5 

358 

402 

155 

19033 

061 

089 

ii7     H5 

173 j  201 

229 

257 

28S 

5  M-5    '4^' 
617.4    r.,X 

il 

1358 

402 

156 

312 

340 

368 

390  1  424 

45'  1  479 

507 

535 

562 

i 

358 

403 

157 

590 

ii% 

645 

673!  700 

728 

756 

783 

811 

838 

720.3    I 

i 

357 

403 

i';S 

86^. 

893 

921 

948     976 

*oo3 

♦030 

*o5il 

♦085 

*II2 

8  23.2    J 
926.1    ^    ,. 

27        iH 

, 

357 

151 
357 

404 

40. 

404 

159 
160 

161 

20  140 

it>7 

194 

222     249 

^6 

303 

330 

385 

H'2 

439 

466 

493     520 

548 

575 

602 

629 

656 

'  683 

710 

'737 

7631  790 

817 

844 

'871 

898 

925 

357 

405 

162 

952 

978 

♦005 

*032  1*059 

♦085 

*iU 

*i39 

♦165 

*I92 

I 

^•7      -•'-■    1 

356 

405 

163 

2Jl,  219 

245 

272 

299 

325 

352 

378 

405 

431 

458 

2 

11      11    i 

356 

400 

164 

"484 

5" 

537 

564 

Sm 

617 

643 

6(69 

696 

722 

3  o»         /"     : 

4  lp.8    ,io.4    1 

513-5    J 30    , 
6  16.2    i;.6    ' 
7li3.9    18.2    ! 

356 

406 

167 

.748 

775 

801 

^-^^ 

880 

906 

932 

958 

985 

356 
356 

406 

407 

22  0ir 

272 

037 

298 

063 
324 

0S9 

350 

"5 

376 

141 
401 

}^ 

'94 

220 

479 

246 
505 

427 

453 

355 

407 

168 

531 

557 

il 

608 

634 

660 

686 

712 

737 

763 

821.6    20.8    i 

r 

255 
355 

408 
'403 
408 

169 
I70 

171 

789 

814 

866 

891 

917 

943 

968 

994 

*oi9 

9124.3  23.4  ! 

25 

23?4|_ 

070. 

096 

121 

147 

172 

198 

223 

249 

274 

'300 

325 

350 

_    

376 

401 

426 

452 

477 

502 

52^8 

\ 

409 

172 

553 

578 

603 

629 

654 

679 

704 

729 

754 

779 

I 

2. 

409 

^n 

805 

830 

851 

880 

905 

930 

9^ 

980 

*oo5 

♦030 

2 

.5- 

410 

174 

24055 

080 

103 

130 

155 

180 

204 

229 

254 

279 

3 
4 

i 

7 

Tn 
TO 

4 

410 

175 

304 

329 

353 

378 

403 

428 

45^ 

477 

502 

527 

12 

15...        ; 

1354 

411 

176 

551 

576 

601 

625 

659 

674 

699 

724 

748 

773 

'353 

411 

177 

797 

822 

846 

895 

920 

944 

969 

993 

♦oi8 

7 

353 

411 

178 

25042 

666 

091 

115 

139 

164 

188 

212 

237 

261 

8 

20'               \ 

353 
353 

412 

179 

180 

285 
527 

310 
55J_ 

.314_ 
575 

358 

382 

406 

431 

_45J. 
696 

479 
4JO 

503 

,744 

9 

22 

/ 

600 

624 

648 

J72 

1  353 

413 

181 

768 

792 

816 

840 

864 

888" 

912 

935 

959 

;983 

24                \ 

352 

413 

182 

26007 

031. 

055 

079 

102 

126 

150 

174 

198 

221 

I 

2.4                 . 

;352 

414 

183 

245 

269' 

293 

316 

340 

364 

387 

411 

435     4^8 

2 

4.8              ; 

'|3S» 

414 

184 

482 

505 

529 

553 

576 

600 

623 

647 

670 

694 

3 

7-2               . 
9.6              1 

I352 

415 

185 

717 

741 

764 

788 

811 

834 

858 

881 

905 

928 

4 

■=T 

415 

415 

186 

187 

951 

27  184  I 

975 
207 

998 
231 

*02I 
254 

*045 
277 

*o68 
300 

♦091 
323 

*ii4 

J46 

*i38 
370 

*i6i 
393 

5j4Z.U          ' 

614.4   ^       } 
7116.8-  1         f 
8;i9.2 
9i2i.6 

22 

.  ,1 

416 

18S 

416 

439 

462 

485 

508 

531 

554 

577 

600 

623 

,3S. 

350 
350 

416 

417 
417 

189 
190 

191 

646 

669 

692 

-Zii 

738 

761 

784 

807 

830 

852. 

875 

898 

921; 

944 

967 

989 

*OI2 

*^35_ 

♦058  *o8i 

2^103 

126 

149 

T^ 

194 

.217, 

240 

262 

285    307 

:-:;o 

418 

192 

^330 

353 

375 

39S 

m 

.443 

466 

488 

5"     533 

i|  2.2 

418 

193 

556 

578 

601 

623 

668 

691 

713 

735     758 

2J  4-4 

419 

194 

780 

803 

825 

8^7  i  870 

892 

914 

937 

959    981 
i^     203 

3 

6.6 

419 

195 

29003 

026 

048 

070 

092 

^^5 

137 

'59 

4 

8.8 

<^o 

196 

226 

248 

270 

292 

3M 

336 

358 

380 

403    425 

^ 

197 

667 

469 

491 

513 

535 

5^57 

579 

601 

623!  045 

f  J98 

688 

710 

73-' 

754 

776 

798 

820 

842  1  1863 

7  '■.^■\ 
817^ 

9" 

.  199 

'885 

907 

929 

_95jj  973, 

994 

*oi6 

*o38 

*o6o 

"^m 

:200 

30i«3 

12? 

146 

i6S"'   190 

2tl 

233 

255 

"276 

298 

N. 

L.    0  !     I 

2 

3    1    4 

5 

6 

7 

8       9 

S.'       T.' 

S."     T." 

''  0;  5      373 

0°    2'=   120"  4-68557     557 

.So" 

4.6S  , 

373      373 

0      3=    iSc              557      557 
c      4  c  240             557      558 

0       29   :-.-  1  74p 

0    30  :=  I  Sod 
0    31  -i860 

3:'2    373 

'  '        -3 

0      25  =i:l\'                                         -,58 

0    26  =15                             -,vS 

0    32  =  1920 

0    27  =16.                          .-58 

0    33  =  1980 

0    28  -16^                            3 58 

0    34  =  2040 

(, 

■\ 

1 

N. 

L.  0 

1 

2. 

3 

4 

5  i  6 

7  18'^ 

9  ^ 

P.P.    1 

200 

20 1 

30103 

125 

146 

168 

190 

211 

233 

255 

:7( 

298 
5H 

5(9.    01  1 

•320, 

341 

363 

384 

406 

428 

449 

471 

492 

202 

535 

557 

578 

600 

621 

643 

664 

68  s 

707 

728 

^ 

203 

750 

771 

792 

814 

835 

856 

878 

899 

920 

942 

3 

4.4   4.2 

6.6  6.3 
8.8  8.4 

204 

963 

984 

*oo6 

♦027 

*048 

*o69 

♦091 

*II2 

*i33 

*IS4 

205 

31175 

197 

218 

239 

260 

281 

302 

323 

34? 

306 

5 

ii.o  10.5 
13.2  12.6 

2o6- 

•387 

408 

429 

450 

471 

492 

513 

534 

555 

576 

6 

207 

597 

618 

639 

660 

681 

702 

723 

744 

7<5? 

78s 

7 

15.4  14.7 

208 

806 

827 

848 

869 

890 

911 

931 

952 

973 

994 

8 

17.6  16.8 

209 
2ro 

211 

3201.5 

035 

056 

077 

098 

118 

139 

160 

181 

201 

9 

19.8  18.9 

222 
428 

243 

263 

284 

305 

325 

346 

366 

387 

408 

on   II 

449 

469 

490 

Sio 

531 

552 

57^ 

593   613 

212 

^34 

654 

675 

69s 

71S 

736 

756 

777 

797   818 

213 

838 

858 

^11- 

899 

919 

940 

960 

980 

*OOI   *021 

4.0 
60 

214 

33041 

062 

082 

102 

122 

143 

163 

183 

203    224 

4 

8.0 

215 

•244 

264 

284 

304 

325 

345 

365 

385 

405    425 

10.0 

216 

445 

465 

486 

506 

52b 

546 

56b 

58b 

606    626 

12,0 

217 

..646 

666 

686 

706 

726 

746 

766 

786 

806    826 

7 

14.0 

2l8 

..846 

866 

885 

905 

925 

945 

965 

985 

*oo5 

*025 

8 

16.0 

219 

220 

221 

34044 

064 

084 

104 

124 

143 

163 

183 

203 

223 

9 

18.0 

•  242 

262 

282 

301 

321 

341 

361 

380 

400 

420 

.0   II 

439 

459 

479 

498 

518 

537 

557 

577 

S96 

616 

222 

•  635 

6S5 

674 

694 

713 

733 

753 

772 

792 

811 

^ 

1.9 
3-8 

9.5 
11.4 

223 

830 

850 

869 

889 

908 

928 

947 

967 

986 

♦005 

3 
4 

224 

35025 

044 

064 

083 

102 

122 

141 

160 

180 

199 

225 

218 

23« 

257 

276 

295 

315 

334 

353 

372 

392 

226 

411 

430 

449 

468 

488 

507 

52b 

545 

564 

583. 

227 

603 

622 

641 

660 

679 

698 

717 

736 

755 

774 

7 

133 

22S 

793 

«I3 

«32 

851 

870 

889 

908 

927 

946 

965 

8 

15.2 

229 
230 

-231 

984 

♦003 

*02I 

♦040 

*o59 

♦078 

*097 

*ii6 

*i35 

*i54 

9 

17.1 

36173 

lyz 

211 

229 

248 

267 

286 

305 

324 

342 

ia   II 

361 

380 

399 

418 

43(? 

455 

474 

493 

511 

530 

3-e 
SA 
7.2 
9.0 
10.8 

232 

549 

S68 

S86 

60s 

62^ 

642 

661 

680 

698 

717 

^ 

2S3 

n^ 

754 

773 

791 

810 

829 

847 

866 

884 

903 

3 
4 

5 
0 

234 

922 

940 

959 

977 

996 

*oi4 

*033 

*osi 

♦070 

*oS8 

235 

37107 

125 

144 

162 

i8i 

199 

218 

236 

2S4 

273 

236 

291 

310 

328 

346 

365 

385 

401 

420 

438 

457 

237 

475 

493 

511 

530 

548 

566 

585 

603 

621 

639 

7 

12.6 

23^S 

658 

676 

694 

712 

731 

749 

767 

785 

803 

822 

8. 

14.4   , 

239 
240 

241 

840 

858 

876 

894 

912 

931 

949 

967 

985 

*oo3 

.  9 

16.2   t 

38021 
202 

039 
220 

057 
238 

075 

o?3 

112 

310 

—148 
328 

166  i  184 

17 

256 

274 

292 

346 

364 

242 

382 

399 

417 

435 

453 

471 

489 

507 

525 

543 

I 

1.7     ;i 

243 

561 

578 

596 

614 

632 

650 

668 

8^3 

703 

721 

2 

3-4 

244 

739 

757 

775 

792 

810 

828 

846  j 

•881 

899 

3 

S.I 

6.8 
8.5 

245 

917 

934 

952 

970 

987 

♦005 

*023  1 

*04i 

♦058  ^076 

4 

24b 

39  ^-94 

III 

129 

146 

164 

182 

199  1 

217 

235  1  252 

I 

7 
8 

247 

270 

287 

305 

322' 

340 

358 

375  i 

393 

410  j  428 

11.9 
i::.6 

1  24S 

445 

463 

480 

498 

515 

533 

550 

568 

585.}  602 

1  249 
I25O 

.^020 

t\^7 

655 

672 

690 

'707 

724 

742 

759  !  777 

.   Q 

rc.^ 

794 

8;i  I 

829 

846 

863 

88 1 

898  ; 

915 

933  i  950 

i  N. 

L.  0 

'■  1 

2  !  3 

4  1  5 

6 

7 

.,8  .  1  9 

P.P. 

S.' 

T.' 

S."  T. 

0° '3^  2160''  4.68  557  55c, 

2'  6.46373 

373 

0°  3'=--  I 

80"  4.68  557  557 

3     373 

373 

04=2 
0  5  =^  3 

40     557  55^^ 
00     557  558 

0.  37  =  2220     557  559 

20     372 

2C        372 

373 
373 

0  38=^2280     557  559 

0  33  =  19 

80     55/  '."'^ 

0  39  =  2340     557  559 

i 

0  34  =  20 
0  35  =  21 

40     557,  '■ 
00     557  - 

r-    40  =  2400     557  559 
-  41  =.2460     556  560 

;  ■ ^ 

0  36  =  21 

60     557  v 

^  =  252.)     556  sf 

/ 

N . 

L.  0 

1      ["2" 

3 

4 

"5 

6 

7  .1  8 

9 

P.  P.    "] 

250 

251 

39  794 

811   S29 

_846 

8C.3 

881 

898 

915 
*o88 

♦100 

950 

♦123 

\   18 

967 

"  98s^  *oo2 

♦019 

*o37 

*054" 

♦071 

252 

40  140 

'57   175 

192 

209 

226 

243 

261 

278 

295 

1 

i.» 

253 

312 

329   34^ 

364 

3^^ 

398 

4«5 

432 

449 

466 

2 

3.^ 

254 

483 

500 

518 

535 

552 

56.9 

586 

603 

620 

637 

255 

654 

671 

688 

705 

722 

739 

756 

773 

790 

807 

256 

824 

841 

_§S8 

875 

892 

909 

926 

943 

960 

970 

(j 

10  0 

257 

993 

*oio 

♦027 

*044 

♦06 1 

♦078 

*09? 

♦in 

*I28 

*M5 

I 

12  b 

258 

41  162 

179 

196 

■'212 

229 

246 

263 

280 

296 

3'3 

14.4 

259 
260 

261 

330_^ 

347 

363 

380 

.J^2L 

414 

430 

_447 

4f>4 

481 

9 

16.2 

497* 

514 

531 

547 

_^64 

581 

5.97 

614 

6^1 

647 

17 

664 

681 

697 

714 

731 

747 

764 

780 

797 

814 

262 

830 

847   863 

880 

896 

913 

929 

946 

963 

979 

I 

^•7  ! 

263 

996 

*OI2 

♦029 

*046 

*o62 

♦078 

♦095 

♦ill 

♦127 

♦  144 

.  2 

3-4 

261 

42  i6o 

-  177- 

-193 

210 

22^ 

243 

259 

275 

292. 

308 

3 
4 

'1 

JO. 2, 

261 

325 

341 

357 

374 

390 

406 

423 

439 

455 

472 

266 

488 

504 

521 

537 

553 

570 

586 

602 

619 

635 

I 

.267 

651^ 

--^67 

♦'  684  1  700 

716 

732 

749 

765 

•7S1 

797 

7 

8 

*  J  '^ 

268 

813 

830 

846 

.  862 

4^78 

894 

911 

927 

943 

959 

269 
270 

271 

175_ 

991 

*oo8 

*024 

♦040 

*o56 

*072 

♦088 

♦104 

*120 

9 

I  >•  > 

43  136 

152 

169 

.185 

201 

217 

233 

249 

265  1  281 

.' 

297 

3^3 

329 

345 

361 

377 

393 

409 

425 

441 

272 

457 

473 

489 

505 

521 

537 

553 

569 

584 

600 

I 

I.O 

273 

616 

632 

648 

664 

680 

696 

712 

727 

743 

759 

2 

g 

8:0 

9.6 

274 

775 

791 

807 

823 

838 

854 

870 

.  886 

902 

917 

3 

275 

933 

949 

965 

981 

996 

*OI2 

*028 

♦044 

♦059 

♦075 

4 

276 

44091 

iSZ. 

£22 

138 

»54 

170 

185 

201 

217 

232 

277 

;.f--248 

264 

279 

295 

^1' 

326 

342 

^ 

373 

389 

I 

278 

,,0404 

420 

436 

451 

467 

483 

498 

514 

52.9 

545 

12.8 

279 
280 

281 

"   ^560 

576 

592 

607 

623 

638 

654 

669 

685 

700 

0 

I4.d 

716 

73^ 

747 

762 

778 

793 

809 

824 

•840 

_8J5. 

15   il 

871 

886 

902 

917 

932. 

948 

963 

979 

994 

*OIO 

282 

45  0-5 

040 

056 

071 

086 

102 

117 

^33 

148 

163 

' 

1.5 

283 

179^ 

194 

209 

225 

240 

255 

271: 

286 

3QI 

317 

2 

30 

284 

>     332 

347 

362!  378 

393 

408 

423 

439 

454 

469 

3 

6.0    1 

285 

484 

500 

515 

530 

545 

561 

576 

591 

606 

6.-' 

4 

286 

637 

652 

667 

682 

697 

712 

728 

743 

758 

7: 

/"•5 

287 

788 

803 

818 

834 

849 

864 

879 

894 

909 

92a 

„ 

JP-S 

t_2.0 

288 

939 

954 

969 

984  1  *OQO 

*ois 

♦030 

♦045 

♦060 

*o-/5 

8 

2S9 

290 

291 

46  090 

i05_ 

120 

135 

150 

165 

180 

195 

210 

22? 

'~^.'; 

3^ 
389 

255 

270 

285^ 

434 

300 

315 

^^ 

345 

~359' 

_37 
5^^ 

•  ■^ 

404 

419 

449 

464 

479 

494 

509 

292 

538 

553 

568 

583 

598 

613 

627 

642 

657 

672. 

293 

687 

702 

716 

731 

746 

761 

776 

790 

805 

820 

294 

835 

850' 

864 

879 

894 

909 

923 

938 

953 

967 

I 

4.2 
5.6 

7.-0 

^  4 

295 

982 

997 

*OI2 

*026 

*04i 

^056 

♦070 

♦085 

♦100 

*ii4 

4 

296 

47129 

144 

159 

173 

188 

,  202 

2i7 

232 

246 

2'- 

297 

"^276 

290 

305 

319 

334 

349 

3(>3 

37S 

392.!  4. 

298 

422 

436 

451 

465 

480 

494 

5c^9 

524 

5.v^ 

299 
300 

567 

582 

596 

611 

625 

_^|  654 

669 

68, 

2'h 

727   741 

756 

770 

784 

799 

813 

82^L^4_ 

N. 

L.  0 

1   i  2  1  3 

4 

5 

6  1  7 

^PT^ 

P.P.    ' 

S.' 

T.' 

S."  T." 

S."  T."    i 

2' 

646  37: 

373 

0°  4'=  240"  468557  558 

0°  45'=:  2700" 

■  " 

3 

37: 

373 

0  5  =  300     557  558 

0  46  =  2760 

.    L   .   .« 

25 

372 

'  373 

0  41  =  2460    ..'556  560 

0  47  ~.  2820 

26 

372 

373 

0  42  =  2520     556  560 

0  4S  =  2880 

'.  :'                  ^ 

27 

372 

374 

0  43.=  -'!5'8o      556  560 

0  49  =  2940 

y\.  '     i 

30 

37 

374 

0  44  =  26^,0      556  560 

0       50  .=:  3000 

5..''  .  ' 

_ 

0  45  =  ^700     556  sfo 

. 

fNr 

L.  0 

1 

2 

3 

4 

5 

6 

7  1  8  1  9 

P.P. 

300 

30 1 

47712 
857 

727 

741 

756 

770 

784 

799 

813 

828    842 
972    986 

871 

885 

900 

914 

929 

943 

958 

302 

48001 

015 

029 

044 

058 

075 

087 

lOl   II6^^  130 

15 

303 

144 

159 

173 

187 

202 

216 

230 

2441  259,  273 

304 

287 

302 

316 

330 

344 

359 

373 

387   401 

416 

' 

1-5 

305 

430 

444 

458 

473 

487 

501 

515 

530   544 

558 

2 

30 

306 

572 

586 

601 

615 

629 

643 

657 

671  j  686 

700 

3 

4-5 
6.0 

7-5 

9.0 

10.5 

12,0 

13-5 

307 

7H 

728 

742 

756 

770 

785 

799 

813 

827 

841 

4 

5 
5 

308 

«55 

869 

883 

897 

911 

926 

940 

954 

96^ 

982 

309 
310 

3" 

/l96_ 

*OIO 

*024 

*038 

♦052 

*o66 

*o8o 

*094 

*io8 

♦122 

9 

49  136 

150 

164 

178 

"192 

206 

220 

234 

248 

262 

276 

290 

304 

31S 

332 

346 

360 

374 

388 

402 

312 

415 

429 

443 

457 

47.1 

485 

499 

513 

527  1  541 

353 

554 

568 

582 

596 

•iSio 

624 

638 

651 

665  1  679 

3H 

693 

707 

721 

734- 

748 

762 

776 

790 

803 

817 

14 

315 

831 

845 

859 

872 

^k 

900 

914 

927  1  941 

955 

316 

969. 

982 

996 

*OIO 

*024 

*037 

♦051 

*o65  *079 

♦092 

I 

1.4 

317 

50  106 

120 

133 

147 

161 

174 

188 

202  1  215 

229 

2 

2.8 

318 

"  243 

256 

270 

284 

297 

3" 

•325 

338   352 

365 

3 

4.2 

5-6  . 
7.0 

8.4 
9.8 

319 
320 

321 

379 

393 

406 

420 

433 

447 

461 

474   488 

501 

4 

•  5 
6 

7 

5J1. 

529 

542 

556 

569 

583 

596 

610   623 

__637; 

^\ki 

664 

678 

691 

705 

718 

732 

745  i  759 

772 

322 

786 

799 

813 

826 

840 

853 

866 

880  !  893 

907 

8 

II.-C 

323 

920 

934 

947 

961 

974 

987 

♦ooi 

*oi4 

*028 

*04i 

9 

12.6 

324 

51^51 

068 

o&t- 

-095 

108 

121 

131 

148 

162 

175 

325 

188 

202 

215 

228 

242 

255 

268 

282 

295 
428 

308 

326 

322 

335 

348 

362 

375 

388 

402 

415 

441 

327 

455 

468 

481 

495 

508 

521 

534 

548 

561 

574 

13 

32S 

587 

601 

614 

627 

640 

654 

667 

680  i  693 

706 

I 

1.3 

1  329 
330 

33i 

720 

733 

746 

759 

772 

786 

799 

812 

825 

957 

838 

2 
3 
4 

2.6 
3-9 

7.8 
9.1 

851" 

865 

878 

891 

904 

917 

930 

943 

970 

983 

996 

♦009 

*022 

*035 

*048 

*o6i 

*o75  !  *o88  1  *ioi 

332 

52114 

127 

140 

153 

166 

179 

192 

205  1  218   231 

333 

244 

257 

270 

284 

297 

310 

323 

336'  349   362 

7 

334 

375 

388 

401 

414 

427 

440 

453 

466   479   492 

8 

10.4 

335 

504 

517 

530 

543 

556 

569 

582 

595   608   621 

9 

II. 7 

336 

634 

647 

660 

^73 

686 

699 

71.1 

724  1  737  1  750 

337 

763 

776 

789 

802 

815 

827 

840 

853  ■  866  1  879 

33^ 

892 

90? 

917 

930 

943 

5S6- 
■0S4 

969 

982  !  994  *oo7 

339 
340 

341 

53020 

033 

046 

058 

071 

097 

no:  122;  135 

12 

148 

161 

173 

186 

199 

212 

224 

237  .  250 

263 
390 

I 
2 

1.2 

2.4 

n 

6.0 
7.2 
8.4 
9.6 

275 

288 

301 

314 

326 

^S 

352 

3641  377 

342 

403 

415 

428 

441 

453 

466 

479 

491  !  504  1  517 

3 

343 

529 

542 

555 

567 

580 

593 

605 

618  :  631   643 

4 

344 

656 

668 

681 

694 

706 

719 

732 

744    757  !  769 

5 
5 

345 

.782 

794 

807 

820 

832 

845 

857 

870    882 

895 

I 

346 

908 

920 

933 

945 

958 

970 

983 

995  *oq8 

♦020 

I  347 

54033 

045 

058 

070 

083 

095 

108 

12b  1  133 

145 

9 

10.8 

1  ,-;48 

'i^ 

170 

183 

195 

208 

220 

233 

245  i  258  :  270 

:  .349 
350 

283 

_295 

307 

320 

332 

345 

357 

37^  382  1  394 

407 

419 

432 

444 

456 

469 

481 

494   506  j  518 

1  N. 

L.  0 

1 

2 

3 

4   '■'1 

6 

7    8  1  9 

P.  P. 

1 

S.' 

T.' 

S. 

/  "Y  ff 

S."  T." 

I  3' 

6.46  373 

373 

0° 

5'=  3 

00"  4.68557 

558 

0°  54'=  3240"  4 

.68  556  561. 

'.     4 

373 

373 

0 

6-  3 

60     557 

558 

0  55  =  3300 
■  0  56  =  3360 

556  561 

1,  ->.^ 

372 

374 

0  5 

0  -^  30 

00     55^ 

)  561 

372 

374 

0  5 
0  5 

1  =30 

2  =  31 

60     55^ 
20     55^ 

^  561 

)  561 

057=  3420 

0   rS  -  -<«r, 

c6i 
562 
56i 

0  5 

3  =  3^ 

80     55^ 

j  561 

C) 

. 

0  5 

4  =  32 

40     55^ 

>  561 

_ 

N. 

L.  0 

1   1  2 

3  1  4 

5 

6  1 

7  i  8 

9     P.  P.   ll 

350 

351 

54407 
531 

419 

543 

432 

444 

456 

469 

481 

494 

506 

S18 

555 

568 

580 

593 

605 

617 

642 

352 
353 

-4ff 

66j_ 
790 

'£- 

814 

704 
827 

839 

851 

"864" 

^ 

765> 
888 

13 

354 

900 

913 

9il 

937 

949 

962 

974 

986 

998 

*OII 

1 
2 

2:6 

355 

55  023 

035 

0^7 

060 

072 

084 

096 

108 

121 

133 

3 
4 

3-9 
$•2 

356 

H5 

157 

169 

182 

194 

206 

218 

230 

242 

255 

357 

267 

279 

,291 

3<^3 

315 

328 

340 

352 

364 

376 

5 

6.5 

358 

388 

400 

413 

425 

437 

449 

461 

473 

485 

iH 

6 

7.8 

359 
360 

361 

5?9_ 

522 

534 

546 

558 

570 

582 

594 

606 

618 

7 
8 

9 

9.1 
10.4 
11.7 

630 

642 

654 

666 

678 

691 

703 

715 

727 

739 

751 

763 

775 

787 

799 

811 

823 

835 

847 

859 

362 

871 

883 

895 

907 

919 

931 

943 

955 

967 

979 

363 

991 

♦003 

*oif 

♦027 

♦038 

♦050 

♦062 

♦074 

*o86 

*098 

^ 

364 

56  no 

122 

134 

146 

158 

170 

182 

194 

205 

217 

12 

365 

-229^ 

241 

253 

265 

277 

289 

301 

312 

324 

336 

366 

■34^ 

360 

372 

384 

396 

407 

419 

431 

443 

455 

I 

1.2 

367 

467 

478 

490 

502 

5H 

526 

538 

549 

561 

573 

2 

24 
3-6 
4.8 
6.0 

f.2 
8.4 

368 

583 

597 

608 

620 

632 

644 

656 

667 

679 

691 

3 

369 
370 

371 

703 

714 

726 

738 

750 

761 

773 

785 

797 

808 

4 

1 

820 

832 

844 

855 

867 

879 

891 

902 

•  914 

^26 

937 

949 

961 

972 

984 

996 

♦008 

♦019 

♦031 

*043 

S72 

.57054 

066 

078 

089 

lor 

"3 

i£4 

136 

148 

159 

8 

9.6 

373 

171 

183 

194 

206 

217 

229 

2^1 

252 

264 

276 

9 

10.8 

374 

287 

299 

310 

322 

334 

345' 

357 

368 

380 

392 

375 

403 

415 

426 

438 

449 

461 

473 

484 

496 

507 

376 

519 

SiP 

542 

553 

565 

576 

588 

600 

611 

6^  i 

377 

634 

646 

657 

669 

680 

692 

703 

715 

726  1 

11 

378 

749 

761 

772 

784 

795 

807 

818 

830 

841    .2 

1 

1.1 

379. 
380 

381 

864 
978_ 

875 

8^ 

8^8 

910 

921 

933 

944 

955 

907 

2 
3 
4 

2.2 

3-3 

4r4- 

990 

*o6i 

*oi3 

*024 

*035 

♦047 

♦058 

♦070 

*o8i 

58092 

104 

"5 

127 

138 

149 

161 

172 

184 

^^5 

382 
383 

206 
320 

218 
■  331 

229 
343 

240 
354- 

252 
365 

263 

377 

274 
388 

286 

297 

309 
422 

1 
I 

1:1 

399 

410 

384 

433 

444 

456 

467 

478 

490 

501 

512 

524 

535 

385 

546 

557 

569 

580 

591 

602 

614 

625 

636 

647 

9 

9.9 

386 

-659 

670 

681 

692 

704 

715 

726 

737 

749 

760 

387 

771 

782 

794 

805 

816 

827 

838 

850 

86i 

872 

i 

388 

883 

894 

906 

917 

928 

939 

950. 

961 

973 

984 

1 

389 
390 

391 

995 

*oo6 

*oi7 

*028 

♦040 

*o5i 

*o62 

*073 

*o84 

*095 

10    i 

59  106 

118 

129 

140 

151 

162 

173 

184 

195 

207 
318 

I 
2 

,,0 

2.0   ' 

218 

229 

240 

251 

262 

273 

284 

295 

306 

392 

329 

340 

351 

362 

373 

384 

395 

406 

417 

428 

3 

30 

393 

439 

450 

461 

472 

483 

494 

5C56 

517 

528 

539 

4 

4.0 

394 

550 

561  i  572 

583 

594 

605 

616 

627 

638 

649 

5 

50 

39'^ 

.660 

671 

682 

693 

704 

715 

726 

737 

748 

759 

6 

6.0 

396 

770 

780 

791 

802 

813 

824 

835 

846 

857 

868 

7 

7.0 

397 

879 

890 

901 

912 

923 

934 

945 

956 

966 

977 

8 

8.0 

398 

988 

999 

*OIO 

*02I 

*032 

*043 

*054 

*o65 

♦076 

*o86 

0 

1  9.0 

399 
400 

60097 

108 

119 

130 

141 

152 

163 

173 

184 

195 

i 

206  j  217 

228 

239 

249 

260 

^71 

282  j  293 

304 

N. 

L.  0  1   1 

2 

3 

4 

5  1  6     7  1  8  1  9  1    P.P.    ij 

S.'  T.'  II                S."  T." 

S."  T."   j 

/ 

646373  373 

0°  5'=  300"  4-68  557  55« 

1°  ^'-3660"  .ir>s  - 

'   4 

•               -57  558 

I  2  =^  3720  ^ 

35-- 
39 

-,7  558 

3  -  3780  ■  ^ 

M5  562 

4  -  3840  ' 

40 

35  562 

I    5  ^^  3QOO 

ft  >               55  562 

I    6  i:  3960  , 

|j  I  •         ;55  562 

I    7  T-^  402c 

lo 

/- 

N, 

L.  0 

1   1  2 

3  1  4 

1  5 

6 

7  1  8 

9 

P.P. 

400 

401 

60  206 

217 

228 

239 

249 

260 

271 

282 

2^.3 

304 

314 

325 

336 

347 

358 

369 

379 

390 

401 

412 

402 

423 

433 

444 

455 

466 

477 

487 

498 

509 

520 

403 

531 

541 

552 

563 

574 

584 

595 

606 

617 

627 

404 

638 

649 

660 

670 

681 

692 

703 

713 

724 

735 

405 

.  746 

756 

767 

778 

788 

799 

810 

821 

'V3I 

842 

406 

853 

863 

874 

885 

895 

906 

917   927  J  938 

949 

11 

407 

959 

970 

981 

991 

*002 

*oi3 

♦023 

*034 

*04^ 

*0S5 

I 

I.I 

408 

61066 

077 

087 

098 

109 

119 

130 

140 

151 

162 

2 

2.2 

409 
410 

411 

172 

183 

194 

204 

215 

225 

236 

247 

257 

26}? 

4 

3-3 
4.4 

11 

278 

289 

300 

310 

321 

33^ 

342  !  352 

363 

374 

384 

395 

405 

416 

426 

437 

448 

458 

469  !  479 

412 

490 

500 

5" 

521 

532 

542 

553 

563 

574 

584 

y 

7-7 

413 

595 

606 

616 

627 

637 

648 

658    669 

679 

690 

8 

8.8 

414 

700 

711 

721 

731 

742 

752 

763 

Z73 
S78 

784 

794 

9 

9.9 

415 

805 

815 

826 

836 

847 

857 

868 

888 

899 

416 

909 

920 

930 

941 

951 

962 

972 

982 

993 

*oo3 

4'Z 

62014 

024 

034 

045 

055 

066 

076 

086 

097 

107 

418 

118 

128 

138 

149 

159 

170 

180 

190 

20 1 

211 

419 
420 

421 

221 

232 

242 

252 

263 

273 

284  I  294 

304 

3'? 

10 

325 

335 

346 

356 

366 

}77 

387   397 

408 

418 

428 

439 

449 

459 

469 

480 

490  j  500 

511 

S2I 

422.. 

531 

542 

5S2 

562 

572 

583 

593  1  603 

613 

624 

^ 

1.0 

423 

634 

644 

651 

665 

675 

685 

696  1  706 

716 

726 

2 

2.0 

424 

737 

747 

757 

7(>7 

778 

788 

79S 

808 

818 

829 

3 
4 

3-0 
4.0 

5-0 
6.0 

425 

839 

849 

859 

870 

880 

890 

900 

910 

921 

931 

426 

941 

951 

961 

972 

982 

992 

*002 

♦012 

*022 

*033 

427 

63043 

053 

065 

073 

083 

094 

104 

114 

124 

134 

7 

7.0 

428 

144 

155 

I6S 

175 

i8s 

195 

205 

215 

22s 

230 

8 

8.0 

429 
430 

431 

246 

256 

266 

276 

286 

296 

306 

3»7 

•327 

337 

9 

9.0 

347 

357 

367 

377 

387 

397 

407 

417 
"5A8 

428 

5^ 

438^ 

538 

448 

4S8 

468 

478 

488 

498 

508 

432 

548 

558 

S68 

579 

589 

599 

609 

619 

629 

639 

433 

649 

659 

669 

679 

689 

699 

709 

719 

729 

739 

434 

749 

759 

769 

779 

789 

799 

809 

819 

829 

839 

435 

849 

859 

869 

879 

889 

899 

909 

919 

929 

939 

43t>. 

.--_^41 

959 

969 

979 

988 

998 

*oo8 

*oi8 

♦028 

*038 

437 

64048 

058 

068 

078 

088 

098 

108 

118 

128 

^37 

9 

438 

H7 

157 

167 

177 

187 

197 

207 

217 

227- 

^237 

I 

°-9 

439 
440 

441 

246 

256 

266 

276 

286 

296 

306 

316 

326 

335 

2 
3 

•4 

1.8 

2.7 
3.6 
•4.5 
5-4 
6.3 

345 

355 

365 

375 

385 

395 

404 

414 

424 

434 

444 

454 

464 

473 

483 

493 

503 

5«3 

523 

532 

442 

542 

552 

S62 

572 

582 

591 

6oi 

611 

621 

631 

443 

640 

650 

660 

670 

680 

689 

699 

709 

719 

729 

7 

444 

738 

748 

758 

768 

777 

787 

797 

807 

816 

826 

8 

l-.l' 

445 

836 

846 

8s6 

865 

^7^ 

88^ 

895 

904 

914 

924 

9 

446 

933 

943 

953 

963 

972 

982 

992 

-*002 

♦on  1  *02I 

447 

65031 

040 

050 

060 

070 

079 

089 

099 

108  1   118 

448 

128 

137 

147 

157 

16.7 

176 

186 

196 

205  j   215 

449 
450 

225 

234 

244 

254 

263 

273 

283 

292 

302  i   312 

321 

331 

341   350 

360 

369 

379 

389 

398  1  408 

N. 

L.  0 

1   '  2  !  3  1 

4 

5  1 

6  1  7 

8 

9  1 

P.P. 

S.' 

T.' 

S.' 

T." 

S/'  T." 

4' 

646  373 

373 

0°  6'^    360"  4 

68557 

558 

I^'  9'=  4140"  4 

•68555  563 

5 

373 

373 

0  7  =  420 
0  8  =  480 

557 
557 

558 
558 

I   10  =  4200 
I   11=  4260 
I   12  =  4320 

554  563 
554  564 
554  564 

40 
42 

372 
372 

375 
375 

I  6  =  3960 

555 

563 

43 

371 

375 

I  7  =  4020 

55^ 

563 

I   13=4380 

554  564 

44 

371 

375 

I  8  =  4080 

553 

563 

I  14  =  44  ^.0 

554  564 

45 

371 

375 

I   9  =r  4140 

555 

563 

I   !  ' 

554  564 

u 

1 1 

N. 

L.  0 

1 

k 

3 

4 

5 

6 

7 

8 

9 

P.P. 

450 

451 

65321 
418 

_33i 

427 

_  341 

437 

_35o 
447 

360 

456 

369 
466 

379 

389 

398 

408 
504 

475 

485 

495 

452 

514 

523 

\:>?> 

543 

S52 

562 

571 

581 

591 

600 

453 

610 

619 

029 

639 

048 

658 

667 

677 

686 

696 

454 

706 

7»5 

72? 

734 

744 

753 

763 

772 

782 

792 

4S5 

801 

811 

820 

830 

839 

849 

858 

868 

877 

887 

456 

896 

906 

9.6 

925 

935 

944 

954 

963 

973 

982 

10 

457 

992 

*OOI 

♦oil 

*020 

♦030 

*039 

*049 

♦058 

*o68 

*o77 

I   I.O 

458 

66087 

..  096 

106 

"5 

124 

134 

143 

.  >53 

162 

172 

2 

2.0 

459 
460 

461 

181 

191 

200 

210 

219 

229 

238 

247 

257 

266 

3 

4 

3.0 
4.0 

276 

285 

291 

304 

314 

,323 

332 

342 

351 

361 

46^ 

38a 

389 

398 

408 

417 

427 

436 

445 

455 

462 

-474 

483 

492 

502 

5'i 

521 

530 

539 

549 

7 

7.0 

463 

558 

5^7 

577 

586 

596 

605 

614 

624 

633 

642 

8 

8.0 

464 

652 

661 

671 

680 

689 

699 

708 

717 

727 

736 

9 

9.0 

46  s 

745 

75? 

764 

773 

783 

792 

801 

811 

820 

829 

466 

839 

848 

857 

867 

876 

885 

894 

904 

913 

922 

467 

932 

941 

9  so 

960 

969 

978 

987 

997 

*oo6 

*oi5 

468 

67025 

034 

043 

052 

062 

071 

080 

089 

099 

108 

469 
470 

471 

117 

127 

136 

H5 

154 

164 

173 

182 

191 

201 

9 

210 

219 

228 

237 

247 

256 

265 

274 

284 

293 

302 

3" 

321 

330 

339 

348 

357 

367 

376 

385 

472 

394 

403 

413 

422 

431 

440 

449 

459 

468 

477 

I  1  0.9 

473 

486 

495 

504 

5M 

523 

532 

541 

550 

560 

569 

2  '  " 

i.e 

474 

S78 

S87 

596 

605 

614 

624 

^33 

642 

651 

660 

3 

2.7 
3.6 
4.5 
5.4 
6.^ 

475 

669 

679 

688 

697 

706 

715 

724 

733 

742 

7S2 

4 

476 

761 

770 

779 

788 

797 

806 

815 

825 

834 

843 

I 

'477 

8S2 

861 

870 

879 

888 

^tl 

906 

916 

925 

934 

s^ 

^47^ 

■  943 

952 

961 

970 

979 

988 

997 

*oo6 

*oi5 

*024 

8  7.2 

'  479 
^80 

^81 

68034 

043 

052 

061 

070 

079 

088 

097 

106 

"5 

9  8.1 

124 

133 

142 

151 

160 

169 

178 

187 

196 

205 

215 

224 

233 

242 

251 

260 

269 

278 

287 

296 

482 

305 

314 

323 

332 

341 

350 

359 

368 

377 

386 

4S3 

395 

404 

413 

422 

-431 

440 

449 

458 

467 

476 

484 

485 

494 

502 

5" 

520 

529 

538 

547 

556. 

565 

.85 

574 

S83 

592 

601 

610 

619 

628 

637 

646 

655 

486 

664 

673 

681 

690 

699 

708 

717 

726 

735 

744 

8 

487 

753 

762 

771 

780 

789 

797 

806 

815 

824 

833 

4.S8 

'     842 

8si 

860 

869 

878 

886 

^95 

904 

913 

922 

I  0.8 

489 
490 

491 

931 

940 

949 

958 

966 

975 

984 

993 

*002 

*OII 

2  1.6 

3  24 
4.3.2 

5  |40 

6  i  4.8 

69020 

028 

037 

046 

055 

064 

073 

082 

099 

099 

108 

117 

126 

135 

144 

152 

161 

170 

179 

188 

492 

197 

205 

214 

223 

232 

241 

249 

258 

267 

276 

493 

285 

294 

302 

3" 

320 

329 

338 

346 

355 

364 

7  5-6 

494 

373 

381 

390 

399 

408 

417 

425 

434 

443 

452 

8  6.4 

49  S 

461 

469 

478 

487 

496 

504 

5^3 

522 

531   539 

9  1  7-2 

496 

548 

557 

566 

574 

583 

592 

601 

609 

618   627 

* 

497 

636 

644 

653 

662 

671 

679 

688 

697 

705   7H 

40H 

723 

732 

740 

749 

758 

767 

775 

784 

793   801 

499 
500 

810 

819 

827 

836 

845 

854 

862 

871 

880   888 

! 

897 

906 

914 

923 

932 

940 

949 

958 

966   975 

N. 

|L.  0 

1   1  2  I  3 

4 

5 

6 

7 

8  1  9 

P.  P. 

S.' 

T.' 

S.' 

f     Y." 

S.'  T." 

4' 

646  373 

373 

0°  7'=  A 

20"  4.68  555 

'      558 

1°  1 8' =4680"  4 

.68554  565 

5 

37: 

;   373 

0      8=4 
0  9=5 

80      55/ 
40     55/ 

'      558 
'      558 

I  19  =4740 
I  20  =4800 
I  21  =4860 

554  565 
554  5«^5 
553  566 

45 
48 

37' 
37' 

375 

375 

I  15  =45 

00     55^ 

^  564 

49 

yn 

376 

I  16=45 

60     55^ 

^  565 

I  22  =4920 

553  506 

50 

37] 

376 

I  17  =46 

20'     SSA 

^  565 

I  23=4980 

553  566 

I  18  =46 

80.     55^ 

^  565 

I  24  =  5040 

553  566 

In. 

L.  0 

1   1 

2 

3 

4 

5 

6  1  7 

ii      1   9 

p.p. 

500 

69897 

906 

914 

923 

932 

940 

949  1 

958 

9'>6  1  975 

984 

992 

*OOI 

*OIO 

*oi8 

*027 

♦036  1  *044 

*o,3  ♦062 

502 

70070 

079 

088 

096'' 

105 

114 

122 

131 

ub 

148 

503 

^57 

165 

174 

183 

191 

200 

209 

217 

226 

234 

^04 

243 

252 

260 

269 

278 

286 

29? 

303 

312 

321 

505 

329 

^^ii 

346 

355 

364 

372 

381 

389 

398 

406 

9 

506 

415 

424 

432 

441 

449 

458 

467 

475 

484 

492 

507 

501 

509 

518 

526 

535 

544 

552 

561 

569 

s78 

I 

0.9   1 

508 

586 

603 

612 

621 

629. 

638 

646 

6S5 

603 

2 

1.8 

509 
5fO 

5" 

672 

680 

689 

697 

706 

7H 

7?3 

731 

740 

749 
834 

3 
4 
5 
6 

2.7 
3-6 
4.5 
5-4 

757 

766 

774 

783 

791 

800 

808 

817 

825 

842 

851 

859 

868 

876 

885 

893 

,902 

910 

919 

512 

927 

935 

944 

952 

961 

969 

978 

,986 

99? 

*oo3 

7 

6-3 

513 

71  012 

020 

029 

037 

046 

054 

063 

071 

079 

088 

8 

7.2 

514 

096 

105 

lU 

122 

130 

139 

147 

155 

164 

172 

9 

8.r 

5^5 

181 

189 

198 

206 

214 

223 

231 

240 

248 

257 

51(5 

265 

273 

282 

290 

299 

307 

315 

324 

332 

341 

517 

349 

3S7 

366 

374 

383 

391 

399 

408 

416 

4'^^ 

518 

433 

441 

450 

458 

466 

47? 

483 

492 

500 

508 

520 

521 

517 

525 

533 

542 

550 

559 

5^7 

575 

584 

592 
675 

B 

600 

609 

617 

625 

634 

642 

650 

659 

667 

684 

692 

700 

709 

717 

725 

734 

742 

750 

759 

522 

767 

775 

784 

792 

800 

809 

817 

82s 

834   842 i 

' 

523 

850- 

858 

867 

875 

883 

892 

900 

908 

917 

925 

2 

3 
4 

2-1 

3 '4 
4.0 
4.8 

524 

933 

941 

950 

958 

966 

97? 

983 

991 

999 

*oo8 

S2.S 

2^016 

024 

032 

041 

049 

057 

066 

074 

082 

090 

i 

526 

099 

107 

"5 

123 

132 

140 

148 

156 

165 

173 

527 

\i8i 
\263 

189 

198 

206 

214 

222 

230 

239 

247 

2SS 

7 

5-6 

S28 

272 

280 

288 

296 

304 

313 

321 

329 

337 

S 

64     .; 

529 
530 

1  "531 

'346 
428 

354 

362 

370 

378 

387 

39? 

403 

411 

419 

501 

583 

^1 

436 

444 

452 

460 

469 

477 

48? 

493 

509 

.518 

526 

534 

542 

550 

558 

567 

575 

1  S32 

591 

599 

607 

616 

624 

632 

640 

648 

656 

06^ 

!  533 

673 

681 

689 

697 

705 

713 

722 

730 

738 

746 

1  534 

535 

-154:. 

\K35 

762 
^43 

770 
-85^^ 

•  Ul^ 

l%- 

876 

803 
884 

811 
892 

819 
900 

^ol 

1 , 

536 

916 

925 

933 

941 

949 

957 

965 

973 

981 

989 

537 

997 

*oo6 

♦014 

*022 

*030 

♦038 

*046 

*054 

*062 

*070 

' 

!  53« 

73078 

086 

094 

102 

III 

119 

127 

135 

143 

151 

1 

539 
540 

541 

159 

167 

175 

183 

191 

199 

207 

215 

223 

231 

\           '       s                 i 

239 

247 

255 

263 

272 

280 

288 

296 

304 

312 

5 
6 

3  'v 

4  ^   i 

320 

328 

336 

344 

352 

360 

368. 

376 

;  384 

392 

542 

400 

408 

416 

424 

432 

440 

448 

4S6 

1  464 

472 

7 

4  ^    1 

543 

480 

488 

496 

504 

512 

520 

528 

536 

544 

552 

s 

:      i 

544 

560 

56^ 

576 

584 

592 

600 

608 

616 

624  632 

r 

545 

640 

648vJ 

656 

664 

672 

679 

687 

695 

703   7" 

1 

546 

719 

727 

735 

743 

751 

759 

767 

775 

783'  791 

547 

799 

807 

815 

823 

830 

838 

846   854 

!  Mi,   .^70. 

S4H 

878 

886 

894 

L   902 

910 

918 

926   933 

1  9'4i  !  949 

>549 
550 

957 

965 

973 

981 

989 

997 

*oo5  *oi3 

*020 

-  >28 

107 

74036 

044 

052 

060 

068 

076 

084  1  092 

(  099 

i  N. 

L.  0  1   1   1 

2  1  3  i  4  1  5 

1  6  1  7 

i  8  1  9 

p.p. 

1 
1 

S/ 

T/ 

S. 

II     ^-n 

S."  T." 

1     rf 
1     5 

6.46  373 

373 

0°  S'=  4S0"  4.68  55' 

I     558 

jO 

26'  =  5 1 60"  4 

.68553  56. 

6 

373 

373 

09=  540     55' 
0  10  =  600     55 

f     558 
7     558 

-  J 

27  =5J20 

28  =  5280 

29  =  5340 

553  >'  • 
553  5 
553  5  ■ 

50 

55 

371 
371 

376 
376 

I  23  =  498c     55^ 

^  S66 

1 

I  24  =  5040     55^ 

^  =;% 

30  =  5400 

553  5"7 

I  25  =  5100     55. 

7,     06 

31  =  54 &0 

552  5'^S 



I  26  =  5160      55 

3  537 

32  =  55-0 

552  568 

U 

..  |L.  0 

"1 

2 

3 

4  1  5  1  6  1  7 

,  -8  1  9 

P.  P.    1 

650 

74036 

044 

052 

060 

06S 

076 

0S4 

092 

099 

107 
186 

Nsi 

"5 

123 

131 

139 

147 

155 

162 

IfO 

178 

1 552 

194 

202 

210 

218 

225 

233 

241 

249 

257 

26^ 

1 553 

273 

280 

288 

296 

304 

312 

320 

327 

335 

343 

554 

351 

359 

367 

374 

382 

390 

398 

406 

414 

421 

555 

429 

437 

44? 

453 

401 

46S 

476 

484 

492 

500 

i  550 

507 

5^5 

523 

531 

539 

547 

554 

502 

570 

578 

557 

S86 

593 

601 

609 

6,7 

624 

632 

640 

648 

6s6 

i  SS8 

66,s 

67. 

679 

687 

695 

702 

710 

718 

726 

733 

559 
560 

S6i 

741 

749 

757 

764 

772 

780 

788 

796 

803 

811 

8 

819 

827 

834 

842 

850 

858 

865 

873 

881 

889 

896 

904 

912 

920 

927 

93? 

943 

950 

958 

966 

I 

0.8 

S62 

974 

981 

989 

997 

*oo5 

*OI2 

*020 

*028 

*035 

*043 

2 

1.6 

563 

75051 

059 

066 

074 

0S2 

089 

097 

105 

"3 

120 

3 

2.4 

S^H 

128 

136 

143 

151 

159 

166 

174 

182 

189 

197 

4 

3-2 

5^5 

205 

213 

220 

228 

236 

243 

25' 

259 

266 

274 

5 

4.0 

566 

282 

289 

297 

305 

312 

320 

328 

335 

343 

351 

6 

4.8 

S67 

3S8 

366 

374 

381 

389 

397 

404 

412 

420 

427 

7 
8 

5.6 
6.4 

568' 

435 

442 

450 

458 

465 

473 

481 

488 

496 

504 

569 
570 

571 

511 

5>9 

526 

534 

542 

549 

557 

565 

572 

580 

9 

7.2 

587 
664 

595 

603 

610 

686 

618 

626 

633 

641 

648 

656 

67. 

679 

694 

702 

709 

717 

724 

732 

572 

740 

747 

755 

762 

770 

778 

78s 

793 

800 

808 

573 

815 

823 

831 

838 

846 

853 

861 

868 

876 

884 

574 

891 

899 

906 

914 

921 

929 

937 

944 

952 

959 

575 

967 

974 

982 

989 

997 

*oo5 

*OI2 

*020 

♦027 

*o3? 

576 

76042 

05J 

057 

of>? 

072 

080 

087 

095 

103 

no 

577 

118 

125 

133 

140 

148 

155 

163 

170 

178 

185 

S7« 

193 

200 

208 

215 

223 

230 

238 

245 

253 

260 

579 
580 

S8i 

268 

275 

283 

290 

298 

305 

3^3 

320 

328 

335 

7 

343 

350 

358 

365 

373 

380 

388 

395 

403 

410 

418 

425 

433 

440 

448 

455 

462 

470 

477 

4SI 

S«2 

492 

500 

507 

515 

522 

530 

537 

545 

552 

559 

U.7 

583 

-  567 

574 

582 

589 

W 

604 

612 

619 

626 

634 

2 

1:4 

584 

641 

649 

656 

664 

67! 

678 

686 

693 

701 

708 

3 

4 

^   5 

6 

28 

585 

K   716 

723 

730 

738 

745 

753 

760 

.768 

775" 
849 

78z 

f^ 

586 

-790 

797 

805 

812 

819 

827 

834 

842 

856 

587 

864 

871 

879 

886 

893 

901 

908 

916 

923 

930 

7 

4.9 

S88 

938 

945 

953 

960 

967 

975 

982 

989 

997 

*oo4 

8 

5-6 

589 
590 

59  i 

592 

77012 

019 

026 

034 

041 

048 

056 

06^ 

070 

078 

9 

6.3 

.085 

093 

100 

107 

115 

122 

129 

^37 

144 

151 

159 
.  232 

166 
240 

'^ 

181 

254 

188 
262 

195 

269 

TA 

210 

217 
291 

225 
298 

283 ' 

593 

305 

3^3 

320 

327 

335 

34^ 

349 

357 

364 

371 

594 

379 

386 

393 

401 

408 

415 

422 

430 

437 

444 

595 

452 

459 

466 

474 

481 

488 

495 

503 

510 

517 

596 

525 

532 

539 

146 

554 

561 

568 

576 

583 

590 

597 

597 

605 

612 

619 

627 

634 

64! 

648 

656 

663 

598 

670 

677 

685 

692 

699 

706 

714 

721 

728 

735 

599 
600 

743 

7?o 

757 

764 

772 

779 

786 

793 

801 

80S 

815 

822 

830 

837 

844 

851 

859 

866 

873 

880 

N.  1 

L.  0 

1    ! 

2    3 

4 

5 

'6  1  7 

8 

9 

P 

.p. 

S.' 

TJ 

S."  T." 

6' 

646  37: 

373 

0°  9'=  5 
0  10  =  6 

40"  4-68557  558 
00     557  558 

i°3 
I  3 

5'=  5700"  4.f 

55 
56 

371 
371 

376 
376 

6  =  5760 

I  31  =54 

60     552  568 

I  3 

7  =  5820 

57 

371 

377 

I  32  =  55 

20      552  568 

A  3 

8  =  5880 

58 

371 

377 

I  33  =  55 

80     552  568 

I  3 

9  =  5940 

59 

37c 

377 

I  34  =  56 

40     552  568 

I  4 

0  —  6000 

60 

37c 

>  377 

I  35  =  57 

00     552  569 

'4 

N. 

L.  0| 

1 

2         3  ! 

4.   5 

6 

Me 

9 

P.P. 

600 

77815 

S22 

830 

837 

844 

851 

859 

__866  1  g73_ 

880 

6oi 

887 

895" 

902 

909 

916 

924 

931 

93^   945 

952 

602 

960 

967 

974 

981 

988 

906  1 

*oo3 

♦oio  *oi7 
082  ^o§9 

*025 

603 

78032 

039 

046 

053 

061 

068 

075 

097 

604 

104 

III 

n8 

125 

132 

140 

147 

154   161 

168 

605 

176 

183 

190 

197 

204 

211 

219 

226   233 

240 

8 

606 

247 

254 

262 

269 

276 

283 

290 

297   305 

312 

607 

319 

326 

333 

340 

347 

351 

362 

369   376 

383 

I 

0.8 
1.6 

608 

390 

398 

405 

412 

419 

426 

433 

440 

447 

455 

2 

609 
610 

462 

469 

_47A 

483 

490 

497 

504 

512 

519 

•526 

3 

4 

5 
6 

24 

533 

540 

547 

554 

561 

569 

576 

583 

590 

597 

3-2 
4.0 
4.8 

6u 

604 

611 

618 

625 

633 

640 

647 

654 

661 

668 

612 

675 

682 

689 

696 

704 

711 

718 

725 

732 

739 

7 

5.6 

6.4 

613 

746 

753 

760 

767 

774 

781 

789 

796 

803 

810 

8 

614 

888 

824 

831 

838 

845 

852 

859 

866 

873 

880 

9 

7.2 

615 

893 

902 

909 

916 

923 

930 

937 

944 

951 

616 

958 

965 

972 

979 

986 

993 

*ooo 

*oo7 

♦014 

*02I 

617 

79029 

036. 

043 

ojo 

057 

064 

071 

078 

08I 

092 

618 

099 

106 

i"^ 

120; 

127 

134 

141 

148 

155 

162 

'  ^119 

169 

176 

183 

190 

197 

204 

211 

2l3 

225 

232 

1  026 

239 

246 

253 

260 

267 

274 

~isr 

288 

295 

302 

7 

1  621 

309 

316 

323 

330 

337 

344 

351 

358 

365 

372 

J 

0.7 
1.4 
2.1 

622 

•379 

386 

393 

400 

407 

414 

421 

428 

43? 

442 

2 

3 

4 

623 

449 

456 

463 

470 

477. 

484 

491 

498 

505 

5" 

624 

518 

525 

532 

.539 

546 

553 

560 

567 

574 

581 

^.8 

625 

588 

595 

602 

609 

616 

623 

630 

637 

644 

650 

5 

3-5 

626 

657 

664 

671 

678 

685 

692 

699 

706 

713 

720 

6 

4.2 

627 

727- 

734 

741 

748 

P4 

761 

768 

775 

782 

789 

7 

4-9^ 

628 

796 

803   810 

817 

831 

837 

844 

851 

858 

8 

5-6 

629 
630 

1  631 

865 

872   879 

886 

893 

900 

906 

913 

920 

927 

9 

6.3 

934. 

941  i  948 

955 

962 

"969" 

975 

982 

"989" 

.996 

80003 

010 

017 

024 

030 

037 

044 

051 

058 

065 

6^2 

072 

079 

085 

092 

099 

106 

'o^- 

120 

127 

134 

633. 

147 

154 

161 

168 

175 

182 

J  88 

195 

202 

i  634 

209 

216 

223 

229 

236 

243 

250 

'  257 

264 

271. 

!  635 

277 

284 

291 

298 

30? 

3-12 

^i^ 

325 

332 

339 

636 

346 

353 

359 

366 

373 

380 

387 

393 

400 

407 

6 

<J37 

414 

421 

428 

4^4 

441 

448 

455 

462 

468 

475 

I 

0.6 

638 

•482 

489 

496 

502 

509 

5i^ 

523 

530 

536 

543 

2 

1.2 

639 
640 

641 

618 

557 

564 
632 

570 
638 

577 

584 

591 

598 

604 

611 

3 

4 

1.8 

645 

_^5A 

659 

665 

672 

679 

2.4 

686 

693 

699 

706 

713 

720 

726 

733 

740 

747 

3-0 
-  6 

1  /- . , 

-ci 

r-bO 

767 

774 

781 

787 

794 

801 

808  1  814 

4.2 
4.8 

'35 

841 

848 

855 

862 

868 

875]  882 

02 

909 

916 

922 

929 

936 

943  1  949 

9 

T.4 

:.69 

976 

983 

.990 

996 

♦003 

*oio  I  *oi7, 
077   SS4' 

•  144!  15^ 

.-  ^  ,  -'37 
1  097  1  104 

•  043 
Jii 

050 
117 

057 
124 

064 
131 

070 
137 

1    Ov>0 

j- 

! 

I'.,;  171 

178 

184 

191 

198 

204 

211  !  218 

1 

'3» 

24? 

251 

258 

?65 

271 

278  1  285 

.305 

3" 

3i8 

325 

33^ 

338 

345   351 

0 

■  1  i  2  i  3 

I  4 

5 

!  6  !  7 

1  8  j  9 

P.P. 

■r  ■' 

S.' 

'  T." 

S."  T." 

-;, 

0°  io'=  ( 

)Oo"  4.6855 

J      558 

i°44'=r624o"  . 

i-68  551  571 

- 

0  II  =  ( 

j6o     55' 

7      558 

I  45  ==6300 
"  I  46  =6360 

551  571 
551  571 

I  40  =6c 

xx)     55 

f   570 

I  41  =6c 

>6o      55 

f   570 

I  47  =6420 

550  572 

I  42  =6] 

20     55 

I   570 

I  48  =6480 

550  572 

i 

I  43=6 
I  44=62 

80     55 
540     55 

I      570 
I   571 

I  49  =6540 

550  572 

1  N. 

L.  0 

1 

2 

^  3  14' 

5 

'6^1  1 

8  J 

9 

P.>. 

750 

87  506 

512 

518 

523  1  529] 

535 

541 

547 

552 

558 

'  751 

5^ 

.  570 

576 

581',  5871 

';93 

599 

604 

610 

616 

■v2 

622 

628 

633 

(^39 

64^ 

6,Si 

656 

662 

668 

674 

s.^ 

»  679 

685 

6^1 

697 

703 

708 

7H 

720 

726 

73 » 

754 

737 

743 

749 

754 

760 

766 

772 

777 

783 

789 

755 

79? 

800 

800 

812 

818 

823 

829 

835 

841 

84b 

75^^ 

852 

858 

864 

809 

875 

88 1 

887 

892 

898 

904 

757 

910 

9«5 

9211 
978 

927 

933 

938 

944 

950 

955 

961 

7S8 

967 

973 

984 

990 

996 

*ooi 

♦007 

♦013 

*oi8 

759 
760 

761 

88024 

030 

036 

041 

047 

053 

058 
116 

064 

~121 

070 

076 

R 

081 

087 

093 

098 

104 

no 

127 

^33 
190 

138 

144 

150 

156 

161 

167 

»73 

178 

184 

06 

762 

195 

201 

207 

213 

218 

224 

230 

235 

241 

247 

/> 

1.2 
1.8 

1(^}> 

252 

258 

264 

270 

275 

281 

287 

292 

298 

304 

3. 
4 

764 

309 

315 

3?i 

326 

332 

338 

343 

349 

355 

360 

2.4 

7^>5 

366 

372 

377 

383 

389 

395 

400 

406 

412 

4»7 

5 

3^0 

7O6 

423 

429 

434 

440 

446 

45 » 

457 

4t)3 

468 

474 

6 

3-6 

767 

480 

485 

491 

497 

502 

508, 

513 

519 

525 

530 

7 

4.2 

768 

S36 

542 

547 

553 

559 

564 

570 

57^ 

581 

587 

8 

4.8   • 

769 
770 

771 

593 
649 

598 
655 

604 

610 

615 

621 

627 

632 

638 

643 

9 

5-4 

660 

666 

672 

677 

683 

689 

694 

700 

705 

711 

717 

722 

728 

734 

739 

745 

75O' 

7S6 

772 

762 

767 

773 

779 

784 

790 

795 

801 

807 

812 

773 

818 

824 

829 

835 

840 

846 

852 

857 

863 

868 

774 

874 

880 

885 

891 

832. 

902 

908 

913 

919 

925 

775 

930 

936 

941 

947 

953 

958 

964 

969 

975 

981 

776 

986 

992 

997 

*oo3 

♦009 

*oi4 

*020 

*025 

♦031 

*037 

777 

89042 

048 

053 

059 

064 

070 

076 

081 

087 

092 

77« 

098 

104 

109 

"5 

120 

126 

131 

^Zl 

143 

148 

779 
780 

781 

154 

159 

.^^5 

170 

17b 

182 

187 

193 

198 

204 

ft 

209 

215 

221 

226 

232 

237 

243 

248 

254 

260 

265 

271 

276 

282 

287 

293 

298 

304 

310 

315 

782 

321 

326 

332 

337 

343 

348 

354 

?><^o^ 

365 

371 

' 

0-5 

I.O 

1.5 
2.0 

l^Z 

376 

382 

387 

393 

398 

404 

409 

415 

421 

426 

3 
4 

784 

432 

437 

443 

448 

454 

459 

465 

470 

476 

481 

7^5 

487 

492 

498 

504 

509 

5'5 

520 

5^6 

531 

537 

1 

2.5 

786 

542 

548 

553 

559 

5^4 

570 

575 

581 

58b 

592 

30 

787 

597 

603 

609 

614 

620 

625 

631 

636 

642 

647 

7 

3.5 

788 

653 

658 

664 

669 

675 

680 

686 

691 

697 

702 

8 

4.0 

789 
790 

791 

70S 

713 

719 

724 

730 

735 

741 

746 

752 

757 

9 

4.5 

763 
818 

768 

774 

77^ 

785 

790 

791 

801 

856 

8g7 

812 

867 

823 

829 

834 

840 

845- 

851 

862 

792 

873 

878 

883 

889 

894 

900 

905. 

911 

916 

922 

793 

927 

933 

938 

944 

949 

955 

960 

966 

971 

972 

794 

982 

988 

993 

998 

♦004 

*oo9 

*oi5 

*020 

*Oi:6 

*03i 

795 

90037 

042 

048 

053 

059 

064 

069 

075 

080 

086 

796 

091 

097 

102 

108 

"3 

119 

124 

129 

135 

140 

797 

146 

ly 

157 

162 

168 

173 

179 

184 

189 

195 

798 

200 

206 

211 

217 

222 

227. 

233 

238 

244 

249 

799 
800 

255 

260 

-266 

271 

276 

282 

287 

293 

298 

304 

309 

3H 

320 

325 

32,^ 

7,2,^^ 

342 

347 

352 

358 

N. 

|L.  0 

1   1  2 

3 

4 

5 

6  i  7.  1  8  j  9 

P.P. 

S.' 

T/ 

S."   T." 

S."  T." 

7' 

646  373 

373 

0° 

12'=   720"  4.6855^7   558 

[3  =  780     5^7  558 
[4  =  840      ^57  558 

2°  8'=  7680"  4-68547  578 

8 

373 

373 

0 

2   9=7740      547  578 

2  10=7800      547  578 

"211  =7860      547  579 

75 
80 

369 
369 

380 
380 

0 

2 

5=7500    /S48  577 

2 

6=7560   ./  548  577 
7  =7620   /   548  577 

2  12=7920      547  579 

2 

2  13=7980      547  579 

2 

8. .7680   .   547  578 

2  14=8040      546  579 

-.. 

L.  0 

1   1  2 

3  1  4 

5  1  6'  1 

7 

8  1  9 

PP. 

800 

8oi 

90309 

3M 

320 

32^ 
380 

:u^ 

33(^ 

342 

347 

352  '    358 

363 

369 

374 

3^5 

3'.- 

396 

401 

407  '  412 

802 

417 

423 

428 

434  i  4J9 

445 

450 

455 

461  ]  466 

803 

472 

477 

482 

488   493 

499 

504 

509 

5»5j  520 

804 

526 

531 

536 

542   547 

553 

558 

563 

569!  574 

805 

.S8o 

585 

S90 

596   601 

607 

612 

617 

623   628 

806 

634 

639: 

644 

650 

655 

660 

666 

671 

677 

682 

807 

687 

693 

698 

703 

700 

7U 

720 

725 

730 

736 

808 

741 

747 

7S2 

757 

763 

768 

m 

779 

784 

789 

809 
810 

811 

795 

800 

806 

811 

816 

822 

832 

838 

843 

6 

849 

854 

859 

865 

870 

87^ 

881 

886 

891  1  897  1 

902 

907 

913 

918 

Q24 

929 

934 

940 

945 

950 

812 

956 

961 

966 

972 

977 

982 

.988 

993 

998 

♦004 

I 

0.6 

«'3 

91  009 

014 

020 

025 

030 

036 

041 

046 

052 

057 

2 

1.2 

814 

062 

068 

073 

078 

0.84 

089 

094 

100 

los 

I  10 

3 

1.8 

8.S 

116 

121 

126 

132 

n7 

142 

148 

J  53 

158 

164 

4 

2.4 

816 

169 

174 

180 

185 

190 

196 

201 

2p6 

212 

217 

^ 

30 
3.6 

4.2 

817 

222 

228 

233 

238   243 1 

249 

.^4 

259 

265 

270 

I 

818 

275 

281 

286 

291 

297 

302 

307 

312 

3i8 

323 

48 

819 
820 

821 

328 

334 

339 

344 

350 

35? 

360 

3^5 

37 « 

37^ 

9 

e;.4. 

38  • 

387 

392 

397 

403 

408 

413 

418 

424 

429 

434 

440 

445 

450 

455 

461 

466 

47» 

477 

482 

822 

487 

492 

498 

503 

.SOS 

5H 

519 

524 

529 

535 

«23 

540 

545 

551 

55(> 

561 

566 

572 

577 

582 

587 

824 

593 

S98 

603 

609 

614 

619 

624 

630 

635 

640 

82s 

645 

651 

6s6 

661 

666 

672 

677 

682 

687 

693 

826 

69S 

703 

709 

714 

719 

■724 

730 

735 

740 

745 

827 

751 

7S6 

761 

819 

772 

777 

782 

787 

793 

79S 

828 

803 

808 

8m 

824 

829 

8S4 

840 

84^ 

850 

829 
830 

831 

855 

861 

866 

871 

876 

882 

887 

892 

897 

903 

95S 

"007 

5 

908 

913 

918 

924 

929 

934 

939 

944 

950 

960 

965 

971 

976 

981 

986 

991 

997 

*,gQ^ 

I 

05 
I.O 

8^2 

92012 

ot8 

023 

028 

033 

03S 

044 

049 

054 

059 

2 

^33 

065 

070 

075 

080 

085 

091 

090 

lOI 

106 

III 

3 

^•5 

sh 

117 

122 

127 

132 

137 

143 

148 

153 

158 

163 

4 

2.0 

83s 

169 

174 

179 

184 

189 

195 

200 

205 

210 

215 

5 

2.5 

836 

221 

226 

231 

236 

241 

247 

252 

257 

262 

267 

6 

30 

sm 

273 

278 

283 

288 

293 

298 

304 

309 

3H 

319 

7 

3-5 

838 

324 

330 

335 

340 

345 

350 

355 

3^1 

366 

371 

8 

4.0 

839 
840 

841 

376 

381 

387 

392 

J97 

402 

407 

412 

418 

423 

9 

45 

428 

433 

438 

443 

449 

454 

459 

464 

469 

474 

480 

48^ 

490 

495 

Soo 

505 

5'* 

516 

521 

526 

842 

531 

536 

S42 

547 

552 

",57 

S62 

5^7 

572 

578 

843 

583 

588 

593 

598 

603 

609 

614 

619 

624 

629 

844 

634 

639 

645 

650 

6S5 

660 

665 

670 

675 

681 

845 

686 

691 

696 

701 

700 

711 

716 

722 

727 

!  732 

846 

737 

742 

747 

V  752 

758 

763 

768 

773 

778 

!   783 

847 

788 

793 

799 

804 

800 

814 

819 

824 

829 

'   834 

848 

840 

84s 

850 

855 

860 

865 

870 

875 

88! 

886 

' 

849 
850 

891 

890 

901 

906 

911 

916 

921 

927 

932 

I  937 

942 

947 

952 

957  1  962 

967 

973 

978 

983!  98^ 

N. 

L.  0 

1   1  2  1  3  1  4 

5 

1  6 

7 

1  8    9 

1    P.P. 

S.' 

T.' 

\   S."  T." 

S.-"    T." 

8' 

6.46  373 

373 

0-13'=  7S0"  i.68557  558 

2^ 

i6'=8i6o"4 

.68  ^a6    580 

9 

373 

373 

0  14=  840      557  558 
015=  900      557  558 

2 
2 

'  2 
2 
2 
2 

17  =  8220 

18  =  8280 

546  580 
546  581 

80 
81 
82 
85 

369 
369 
368 
368 

380 
381 
381 
381 

2  13^=7980     -\47  579 
2  14  =  8040      5:,^  579 
2  15  =8100      54t  580 

2  i6  =  8i'o      546]  5 So 

19=8 

20  =  8 

21  =;:8 
22=8 

340 

400 
460 
520 

54 
54 
54 
54 

t>  5»» 
5  582 
5  582 
5  582 

N. 

L.  0 

1   1  2 

3 

4 

5 

6 

7  1 

8  1 

9 

P.P. 

850 

8si 

92942 

947 

_951 
♦003 

957 
*oo8 

962 
*oi3 

967 
*oi8 

♦024 

978 

983 

988 

993 

998 

♦029 

*034 

*039 

8s2 

93  044 

049 

054 

059 

064 

069 

221 

080 

085 

090 

«53 

095 

100 

105 

no 

i'5 

120 

»25 

13'. 

130 

141 

8^4 

146 

151 

156 

161 

166 

171 

176 

181 

.186 

192 

S55 

197 

202 

207 

212 

217 

222 

227 

232 

237 

242 

a 

856 

247 

252 

258 

263 

268 

273 

278 

283 

288 

293 

06 
I  2 

8S7 

298 

303 

308 

313 

318 

323 

328 

334 

339 

344 

2 

858 

349 

354 

359 

364 

369 

374 

379 

384 

389 

394 

3 
4 

5 
6 

I  s 

859 
830 

So  I 

399 

404 

409 

414 

420 

425 

430 

433 

440 

445 

2.4 
3-0 
3.6 

430 

455 

460 

46? 

470 

475 

480 

485 

490 

495 

Soo 

505 

510 

515 

520 

526 

531 

536 

541 

546 

S62 

551 

556 

S6i 

S66 

571 

576 

581 

586 

591 

59b 

7 

4.2 

86,:; 

601 

bo6 

611 

616 

621 

626 

631 

636 

641 

646 

8 

4.8 

S64 

651 

656 

661 

666 

671 

676 

682 

687 

692 

697 

9 

5-4 

865 

702 

707 

712 

717 

722 

727 

732 

737 

742 

747 

86b 

752 

757 

762 

767 

772 

777 

782 

787 

792 

797 

867 

802 

807 

812 

817 

822 

827 

832 

837 

842 

847 

868 

852 

857 

862 

867 

872 

877 

882 

887 

892 

897 

869 
870 

871 

902 

95^ 
94002 

907 

912 

917 

922 

927 

932 

937 

942 

947 

5 

957 

962 

967 

972 

977 

982 

987 

992 

997 

007 

012 

017 

022 

027 

032 

037 

042 

047 

872 

052 

057 

062 

067 

072 

077 

082 

086 

091 

096 

I 

05 

873 

lOI 

106 

III 

116 

121 

126 

131 

13(5 

141 

146 

2 

I.O 

874 

151 

iS6 

161 

166 

171 

176 

i8i 

186 

191 

196 

3 
4 

1-5 

0  cf 

87s 

201 

2Q6 

211 

216 

221 

226 

231 

236 

240 

245 

2-5 
30 

3-5 

876 

250 

255 

260 

265 

270 

275 

280 

285 

290 

295 

877 

300 

305 

310 

315 

320 

325 

330 

335 

340 

34? 

^ 

878 

349 

354 

359 

364 

3^9 

374 

379 

384 

389 

394 

8 

4.0 

879 
880 

881 

399 

404 

409 

414 

419 

424 

429 

433 

438 

443 
493 

9 

4.5 

448 
498 

453 

458 

463 

468 

473 

478 

483 

488 

503 

507 

512 

517 

522 

527 

5f 

537 

542 

882 

547 

SS2 

557 

562 

567 

571 

57^^ 

581 

586 

591 

883 

596 

601 

606 

611 

616 

621 

626 

630 

^35 

640 

884 

645 

650 

655 

660 

665 

670 

673 

680 

685 

689 

88s 

694 

699 

704 

709 

714 

719 

724 

729 

734 

738 

886 

743 

748 

753 

758 

7^3 

768 

773 

778 

783 

787 

4 

887 

792 

797 

802 

807 

812 

817 

822 

827 

832 

836 

I 

0.4 

888 

841 

846 

851 

856 

861 

m 

871 

876 

880 

885 

2 

0.8 

889 
890 

891 

890 

895 

900 

905 

910 

915 

919 

924 

929 

934 

3 

4 

5 
6 

1.2 
1.6 
2.0 
2.4 
28 

939 

944 

949 

954 

959 

963 

968 

973 

*022 

978 

983 

988 

993 

998 

*002 

♦007 

*OI2 

♦017 

♦027 

*032 

892 

95036 

041 

046 

051 

056 

06 1 

066 

071 

075 

080 

7 
8 

893 

085 

090 

095 

100 

105 

109 

114 

119 

124 

129 

32 
3.6 

894 

134 

139 

143 

148 

153 

158 

163 

168 

173 

177 

9 

89s 

182 

187 

192 

197 

202 

207 

211 

216 

221 

226 

896 

231 

236 

240 

245 

250 

255 

260 

265 

270 

274 

897 

279 

284 

289 

294 

299 

303 

308 

3^3 

318 

323 

898 

328 

332 

337 

342 

347 

352 

357 

3t>i 

36b 

371 

899 
900 

37^ 

381 

386 

390 

395 

400 

405 

410 

415 

419 

424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

n: 

1  L.  0  1   1   !  .  2  1  3 

4 

5 

1  6 

7  1  8 

1  9 

1    P.P. 

S.'  TJ    1 

S."  T." 

S."  T." 

8' 

6.46  373   373 

0°  14'=  h 

>4o"  A 

.68  557  558 

2°25'  =  87 

00"  . 

^68545  583 

9 

373  373 

0  15  -  9 

00 

557   558 

2  26  =87 

60 
20 

544  584 
544  584 

8S 

368  381 

2  21  =84 

^6o 

545  582 

"  2  27  =88 

86 

368  382 

2  22  =81: 

20 

545  582 

2  28=8S 

80 

544  584 

89 

368  382 

2  23  =8c 

80 

545  583 

2  29=89 

40 

544  585 

r 

368  383 

2  24=;8e 

2  25-=8; 

)40 
700 

545  583 
545  583 

2  30=9C 

00 

544  585 

20 

N. 

L.  0|  1 

2 

3 

1  4  1  5  1  6 

7 

8 

9  1    P.P.  ~]\ 

900 

901 

95424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

472 

477 

482 

487 

492 

497 

501 

506 

5" 

516 

902 

521 

525 

530 

535 

540 

54,"; 

SSo 

554 

559 

564 

903 

569 

574 

578 

583 

588 

593 

598 

602 

607 

612 

904 

617 

622 

626 

631 

636 

641 

646 

650 

6S5 

660 

905 

665 

670 

674 

679 

684 

689 

694 

698 

703 

708 

90b 

713 

718 

722 

727 

732 

737 

742 

746 

751 

756 

907 

761 

766 

770 

775 

780 

785 

789 

794 

799 

804 

908 

809 

813 

818 

823 

828 

832 

837 

842 

847 

8S2 

909 
910 

911 

856 

861 

866 

871 

875 

880 

885 

890 

895 

899 

6    1 

904 

909 

914 

918 

923 

928 

933 

938 

942 

947 

952 

957 

961 

966 

971 

976 

980 

985 

990 

995 

0.5 

912 

999 

♦004 

♦009 

*oi4 

♦019 

♦023 

*028 

*033 

*os8 

*042 

2 

I.O 

913 

96047 

052 

057 

061 

066 

071 

076 

080 

085 

090 

3 

1-5 

914 

095 

099 

104 

109 

114 

118 

123 

128 

133 

137 

4 

2.0 

915 

142 

147 

152 

156 

ibi 

166 

171 

175 

180 

185 

5 

2-5 

916 

190 

194 

199 

204 

209 

213 

218 

223 

227 

232 

6 

3.0 

917 

237 

242 

246 

251 

2S6 

261 

26s 

270 

275 

280 

I 

3-5 

918 

284 

289 

294 

298 

303 

308 

313 

317 

322 

327 

4.0 

919 
920 

921 

332 

33(^ 

341 

346 

350 

355 

360 

365 
412 

369 

374 

9 

4.5 

379 

384 

388 

393 

398 

402 

407 

417 

421 

426 

431 

435 

440 

445 

450 

454 

459 

464 

468 

922 

473 

478 

483 

487 

492 

497 

501 

506 

5" 

515 

923 

520 

525 

530 

534 

539 

544 

548 

553 

558 

562 

924 

567 

572 

577 

581 

S86 

S9I 

595 

600 

60s 

609 

925 

614 

619 

624 

628 

633 

638 

642 

647 

6S2 

656 

926 

661 

666 

670 

675 

680 

685 

689 

694 

699 

703 

927 

708 

713 

717 

722 

727 

731 

736 

741 

745 

750 

928 

755 

759 

764 

769 

774 

778 

783 

788 

792 

797 

929 
930 

931 

802 

811 

816 

820 

825 

830 

834 
881 

839 

844 

4 

-S48 

453 

858 

862 

867 

872 

876 

886 

890 

895 

900 

904 

909 

914 

918 

923 

928 

932 

937 

932 

942 

946 

951 

956 

960 

965 

970 

974 

979 

984 

I 

U.4 
0.8 

933 

988 

993 

997 

*002 

♦007 

♦oil 

*oi6 

*02I 

♦025 

♦030 

3 
4 

934 

97035 

039 

044 

049 

053 

058 

063 

067 

072 

077 

1.6 

935 

081 

086 

090 

095 

ICX) 

104 

109 

114 

118 

123 

5 
6 

936 

128 

132 

137 

142 

146 

151 

155 

160 

165 

169 

2.4 

937 

174 

179 

183 

188 

192 

197 

202 

206 

211 

216 

7 

2.8 

93« 

220 

225 

230 

234 

239 

243 

248 

253 

2S7 

262 

8 

3-2 

939 
940 

941 

267 

271 

276 

280 

285 

290 

294 

299 

304 

308 

9 

3-6 

313 

3'7 

322 

327 

331 

336 

340 

345 

350 

354 

359 

364 

368 

373 

377 

382 

387 

391 

39^ 

400 

942 

405 

410 

414 

419 

424 

428 

41^ 

437 

442 

447 

943 

451 

456 

460 

465 

470 

474 

479 

483 

488 

493 

944 

497 

502 

506 

5" 

516 

520 

525 

529 

534 

539 

945 

543 

548 

552 

SS7 

S62 

S66 

571 

575 

580 

585 

946 

589. 

594 

598 

603 

607 

612 

617 

621 

626 

630 

947 

635 

640 

644 

649 

653 

6s8 

663 

667 

672 

676 

948 

681 

685 

690 

695 

699 

704 

708 

713 

717 

722 

949 
950 

727 

731 

736 

740 

745 

749 

754 

759 

763 

768 

772 

777 

782 

786 

791 

795 

800 

804 

809 

813 

N. 

L.  0 

1   1 

2  1 

3  1 

4 

5  1  6  1  7  1 

8  1 

9  1    P.P. 

S/ 

T.' 

S."  T." 

S."  T." 

9' 

6.46  373 

373 

0°  I 

5'-  S 

00"  4-68557  558 

2°  3^ 

^'=92 

^0"  4.68543  587 

10 

373 

373 

0  I 

6=    c 

)6o      557  558 

2  3 
2  3 

5=93 
3  =  93 

30      543  587 
So      543  587 

90 

368 

383 

2  1 

0  =  9C 

)oo      544  585 

91 

368 

383 

2  -i 

I  =9C 

)6o      544  585 

2  3 

7  =94 

20      542  588 

92 

367 

383 

2  3 

2=91 

20      543  586 

2  3 

^  =  94 

So      542  588 
4.0      542  588 

94 

367 

383 

2  3 

3  =  91 

80      543  586 

2  3 

:^=95 

95 

367 

384 

2  3 

4  =  9: 

'AO              543  587 

N.   L.  0  1   1   1 

2  1  3  1  4  1 

5    6  1 

7  1 

8  1 

9 

p.pr 

950 

97  772 

777 

782 

786   791 1 

795 

800 

804 

809 

813 

95' 

818 

823 

827 

832 

836 

841 

845 

850 

855 

859 

952 

864 

868 

873 

877 

882 

886 

891 

896 

900 

905 

9S3 

909 

914 

918 

923 

928 

932 

937 

941 

946 

950 

954 

955 

959 

964 

968 

973 

978 

982 

987 

991 

996 

955 

98000 
•"^046 

00^ 

009 

014 

019 

023 

028 

032 

037 

041 

956 

050 

055 

05,9 

064 

068 

073 

078 

082 

087 

957 

091 

096 

100 

105 

109 

114 

118 

123 

127 

132 

9S8 

137 

141 

146 

150 

155 

159 

164 

168 

173 

177 

959 
980 

961 

182 

i85 

191 

195 

200 

204 

254 

214 

218 

223 

5 

227 
272 

232 

236 

241 

245 

250 

_^59_ 
304 

263 
308 

268 
"~3iT 

277 

281 

286 

290 

295 

299 

962 

3'8 

322 

327 

331 

336 

340 

345 

349 

354 

358 

I 

"•5 

9^^ 
964 

408 

367 
412 

372 
417 

376 
421 

381 
426 

385 
430 

390 
435 

394 
439 

399 
444 

403 
448 

2 
3 
4 

5 

6 

I.O 

1-5 

96s 

453 

457 

462 

466 

471 

475 

480 

484 

489 

493 

2-5 

3.0 

966 

498 

502 

507 

5" 

516 

520 

525 

529 

534 

538 

967 

543 

547 

552 

556 

561 

565 

570 

574 

579 

583 

7 

3.5 

968 

S88 

592 

597 

601 

605 

610 

614 

619 

623 

628 

8 

4.0 

969 
970 

971 

632 

637 

641 

646 

650 

655 

659 

664 

668 

673 

9 

4.5 

677 

682 

686 

691 

695 

700 

704 

709 

7'3 

717 

722 

726 

731 

735 

740 

744 

749 

753 

75^ 

762 

972 

767 

771 

776 

780 

784 

789 

793 

798 

802 

807 

973 

811 

816 

820 

825 

829 

834 

838 

843 

847 

851 

974 

856 

860 

865 

869 

874 

878 

883 

887 

892 

896 

1  975 

900 

905 

909 

914 

«  918 

923 

927 

932 

936 

941 

i  97(3 

945 

949 

954 

958 

963 

967 

972 

976 

981 

985 

1  977 

989 

994 

998 

*oo3 

*oo7 

*OI2 

*oi6 

*02I 

♦025 

♦029 

1  978 

99034 

038 

043 

047 

052 

056 

o6j 

065 

069 

074 

979 
980 

981 

078 

083 

087 

092 

096 

100 

105 

109 

114 

118 

A 

123 

127 

131 

136 

140 

145 

149 

154 

158 

162 

167 

171 

176 

180 

185 

189 

193 

198 

202 

207 

0.4 
08 

q82 

211 

216 

220 

224 

229 

233 

238 

242 

247 

251 

983 

255 

260 

264 

269 

273 

277 

282 

286 

291 

295 

3 
4 

1.2 

984 

300 

304 

308 

3>3 

317 

322 

326 

330 

335 

339 

1.6 

98s 

344 

348 

352 

357 

361 

366 

370 

374 

379 

383 

5 

2.0 

986 

388 

392 

396 

401 

405 

410 

414 

419 

423 

427 

6 

2.4 

9S7 

432 

436 

441 

445 

449 

454 

458 

463 

467 

471 

7 

2.8 

988 

476 

480 

484 

489 

493 

498 

502 

506 

5" 

515 

8 

3-2 

989 

990 

991 

520 

524 

528 

533 

537 

542 

546 

550 

555 

559 

9 

3.6 

564 

568 

572 

577 

581 

,585 

590 

594 

599 

603 

•  607 

612 

616 

621 

62s 

629 

634 

638 

642 

647 

9^2 

651 

656 

660 

664 

669 

673 

677 

682 

686 

691 

993 

695 

699 

704 

708 

712 

717 

721 

726 

730 

734 

994 

739 

743 

747 

752 

756 

760 

765 

769 

774 

778 

995 

782 

787 

791 

795 

800 

804 

808 

813 

817 

822 

996 

82a 

830 

835 

839 

•843 

848 

852 

856 

861 

865 

997 

870 

874 

878 

883 

887 

891 

896 

900 

904 

909 

qq8 

913 

917 

922 

926 

930 

935 

939 

944 

948 

952 

1  999 
lOOC 

957 

961 

965 

970 

974 

978 

983 

987 

991 

996 

00000 

004 

009 

013 

017 

022 

026 

030 

035 

039 

N.  1  L.  0 

1   I 

2  1  3 

4  1  5  1  6  1  7  1  8 

1  9  1    p.p. 

S.'   T.' 

S."  T.' 

S/'  T." 

9'  6.46373   373 

0°  15'=  900"  4.68557  55^' 

2°  41'=  c 

)66o"  4.68542  589 

10     373   373 

0  16  =  960      557  55^ 

2  42=  c 
>   2  43=  c 

)720      541  590 
?78o      541  590 
?840      541  590 

95     367   384 
98     367   384 

0  17=1020      557   55^ 

2  38=9480      542  58^ 

\      2  44  =  < 

99     367   385 

2  39  =9540      542  58^ 

\      2  45  =  < 

J906      541  591 

100     366   385 

2  40  =9600     542  58c 

J   2  46=  < 

^960      541  591 

2  41  =9660     542  58c 

)   2  47=i( 

D020      540  592 

N. 

L.   0 

1 

2 

3   1   4 

5     6 

7 

8 

9 

1000 

lOOI 
I002 

1003 

0000000 

0434 

0869 

1303 

1737 

21 7 1   2605 

3039 

3473 

3907 

4341 

8677 

001  3009 

4775 
9111 

3442 

5208 
9544 
3875 

5642 

9977 
4308 

6076 
♦0411 

4741 

6510 

*o844 

5174 

^^943 

*I277 

5607 

7377 

*i7io 

6039 

7810 

*2i43 
6472 

8244 

♦2576 

6905 

1004 
1005 
1006 

7337 

002  ibbi 

5980 

7770 
2093 
6411 

8202 

2525 
6843 

8635 
2957 
7275 

9067 

3389 
7706 

9499 
3821 
8138 

9932 
4253 
8569 

♦0364 
4685 
9001 

♦0796 
5116 
9432 

♦1228 
5548 
9863 

1007 
1008 
1009 

iOlO 

lOI  I 

1012 
1013 

003  0295 
4605 
8912 

0726 
5036 
9342 

5467 
9772 

1588 

5898 

♦0203 

2019 

6328 

*o633 

2451 

6759 

♦1063 

2882 

7190 

*I493 

3313 

7620 

♦1924 

3744 

8051 

*2354 

4174 

8481 

♦2784 

0043214 

3644 

4074 

4504 

4933 

5363 

5793 

6223 

6652 

7082 

7512 

005  1805 

6094 

7941 
2234 

6523 

8371 
2663 
6952 

8800 
3092 
7380 

9229 

3521 
7809 

9659 
3950 
8238 

*oo88 

4379 
8666 

♦0517 
4808 
9094 

*0947 
5237 
9523 

*i376 
5666 
9951 

1014 
1015 
1016 

006  0380 
4660 
8937 

0808 
5088 
9365 

1236 
5516 
9792 

1664 

5944 
♦0219 

2092 

6372 

♦0647 

2521 

6799 

*io74 

2949 

7227 

♦1501 

3377 

7655 

*I928 

3805 
8082 

*2355 

4233 

8510 

♦2782 

1017 
1018 
1019 

1020 

1021 
1022 
1023 

0073210 

7478 
008  1 742 

3^37 
7904 
2168 

4064 
8331 
2594 

4490 

8757 
3020 

4917 
9184 
3446 

5344 
9610 

3872 

5771 
*oo37 

4298 

6198 

♦0463 

4724 

6624 

*o889 

5150 

7051 
*i3i6 

5576 

6002 

6427 

6853 

7279 

7704 

8130 

8556 

8981 

9407 

9832 

0090257 

4509 
8756 

0683 

4934 
9181 

1 108 

5359 
9605 

f533 
.5784 
♦0030 

1959 
6208 

*0454 

2384 

6633 

♦0878 

2809 

7058 

♦1303 

3234 

7483 

*I727 

3659 
7907 

*2I5I 

4084 

8332 

*2575 

1024 
1025 
1026 

010  3000 

7239 
on  1474 

3424 
7662 

1897 

3848 
8086 
2320 

4272 
8510 
2743 

4696 

8933 
3166 

5120 

9357 
3590 

5544 
9780- 

4013 

5967 

*0204 

4436 

6391 

*o627 

4859 

6815 
♦1050 

5282 

1027 
1028 
1029 

1030 

1031 
1032 
1033 

5704 

9931 

012  4154 

6127 

*0354 

4576 

6550 

♦0776 

4998 

6973 

*ii98 

5420 

7396 

*I62I 

5842 

7818 

*2043 

6264 

8241 

♦2465 

6685 

S664 

♦2887 

7107 

9086 

*33»o 

7529 

9509 

*3732 

7951 

8372 

8794 

9215 

9637 

*oo59 

*048o 

*090i 

♦1323 

*I744 

*2i65 

0132587 

6797 

014  1003 

3008 
7218 
1424 

3429 
7639 
1844 

3850 
2264 

4271 
8480 
2685 

4692 
8901 
3105 

5"3 
9321 

3525 

5534 
9742 
3945 

5955 
♦0162 

4365 

6376 

*0583 

4785 

1034 
1035 
1036 

5205 

9403 

015  3598 

5625 
9823 
4017 

6045 

*0243 

4436 

6465 

*o662 

4855 

6885 

♦1082 

5274 

7305 
♦1501 

5693 

7725 
♦1920 

6ll2 

8144 

*2340 

6531 

8564 

*2759 

6950 

8984 

♦3178 

7369 

»037 
1038 
1039 

I040 

104 1 
1042 
1043 

7788 

016  1974 

6155 

8206 
2392 
6573 

8625 
2810 
6991 

9044 
3229 
7409 

9462 
3647 
7827 

9881 
4065 
8245 

♦0300 

4483 
8663 

♦0718 
4901 
9080 

*ii37 
5319 
9498 

*i555 
5737 
9916 

0170333 

075  • 

1168 

1586 

2003 

2421 

2838 

3256 

3673 

4090 

4507 

'  8677 

0182843 

4924 
9094 
3259 

5342 

95" 
3676 

5759 
9927 
4092 

6176 

*0344 
4508 

6593 
*076i 

4925 

7010 

*ii77 

5341 

7427 

*i594 

5757 

7844 

*20I0 
6173 

8260 

♦2427 

6589 

1044 

1045 
1046 

7005 

019  1 163 

5317 

7421 
1578 
5732 

7837 
1994 
6147 

8253 
2410 
6562 

8669 
2825 
6977 

9084 
3240 
7392 

9500 
3656 
7807 

9916 
4071 
8222 

*0332 

4486 

8637 

*0747 
4902 

9052 

1047 
1048 
1049 

1050 

,^67 
020  3^r3 

7755 

4027 
8169 

♦0296 
4442 
8583 

♦071 1 
4856 
8997 

*II26 

5270 
941 1 

*i540 
5684 
9824 

*i955 

6099 

♦0238 

♦2369 

6513 
♦0652 

♦2784 
6927 

*io66 

♦3198 

7341 
•*I479 

021  1893 

2307 

2720 

3134 

3547 

3961 

4374 

4787 

5201 

5614 

N. 

L.   0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

S." 

T." 

S."    T." 

2°  46'  =  9960 

'    4. 

58541 

591 

2°5i'  =  10260" 

4.68  540    593 

2  47  =  10020 

540 

592 

2  52  =  10320 

539    594 

2  48  =  10080 

540 

592 

2  53  =  10380 

539    594 

2  49  =  10140 

540 

592 

2  54  =  10440 

539    595 

2  50  =  10200 

540 

593 

2  55  =  10500 

539    595 



23 

N. 

L.   0  1   1 

2 

3   1   4 

5 

6   1   7   1   8 

9 

1050 

1051 
1052 
1053 

021  1893 

2307 

2720 

3»34 

3547 

3961 

4374 

4787 

5201 

5614 

6027 

0220157 

4284 

6440 
0570 
4696 

6854 
0983 
5»09 

7267 
1396 
5521 

7680 
1808 
5933 

8093 
2221 

6345 

8506 
2634 
6758 

8919 
3046 
7170 

9332 
3459 

7582 

9745 
3871 
7994 

»054 

1055 
105b 

8406 

0232525 

6639 

8818 
2936 
7050 

9230 
3348 
7462 

9642 
3759 
7873 

♦0054 
4171 

8284 

♦0466 
4582 
8695 

♦0878 

4994 
9106 

*1289 
5405 
9517 

♦1701 

5817 
9928 

♦2113 

6228 

*0339 

1057 
1058 
1059 

1060 

1061 
1062 
1063 

0240750 

4^57 
8960 

1161 

5267 
9370 

9780 

1982 

6088 

♦0190 

2393   2804 

6498   6909 

♦0600  *IOIO 

3214 

7319 

♦1419 

3625 

7729 

*i829 

4036 

8139 
♦2239 

4446 

8549 
*2649 

025  3059 

3468 

3878 

4288 

4697 

5^07 

5516 
9609 
3698 
7783 

5926 

6335 
*0427 

4515 
8600 

6744 

*o836 

4924 
9008 

7154 
026  1245 

5333 

7563 
1 654 
5741 

7972 
2063 
6150 

8382 
2472 
6558 

8791 
2881 
6967 

9200 
3289 
7375 

*ooi8 
4107 
8192 

1064 
1065 
1066 

9416 
027  3496 

7572 

9824 
3904 
7979 

*0233 
4312 
8387 

♦0641 
47'9 
8794 

*i049 
5127 
9201 

*i457 
5535 
9609 

♦1865 

5942 

*ooi6 

♦2273 

6350 
♦0423 

*268o 

6757 
♦0830 

*3o88 

7165 

*i237 

1067 
1068 
1069 

I070 

1071 
1072 
1073 

028  1644 
5713 
9777 

2051 

6119 

*oi83 

2458 

6526 

*059o 

2865 

6932 

♦0996 

3272 

7339 
♦1402 

3679 

7745 

*i8o8 

4086 

8152 

*22I4 

4492 
♦2620 

4899 

8964 

♦3026 

5306 

9371 

*3432 

029  3838 

4244 

4649 

5055 

5461 

5867 

6272 

6678 

7084 

7489 

7«95 
030  1948 

5997 

8300 

2353 
6402 

8706 
2758 
6807 

91U 
7211 

9516 
3568 
7616 

9922 

3973 
8020 

♦0327 
4378 
8425 

*0732 

4783 
8830 

*ii38 
5188 
9234 

*i543 
5592 
9638 

1074 

1075 
1076 

031  0043 
4085 
8123 

0447 
4489 
8526 

0851 
4893 
8930 

1256 
5296 
9333 

1660 
5700 
9737 

2064 

6104 

*oi40 

2468 
6508 

*0544 

2872 

6912 

♦0947 

3277 

7315 

*i350 

3681 

7719 

*i754 

1077 
1078 
1079 

1080 

1081 
1082 
1083 

0322157 

6188 

0330214 

2560 
6590 
0617 

2963 

6993 
1019 

3367 
7396 
1422 

3770 
7799 
1824 

4173 
8201 
2226 

4576 
8604 
2629 

4979 
9007 

3031 

5382 
9409 
3433 

5785 
9812 

3835 

4238 

4640 

5042 

5444 

5846 

6248 

6650 

7052 

*io68 

5081 

9091 

7453 

♦1470 

5482 

9491 

7855 

8257 

034  2273 

6285 

8659 
2674 
6686 

9060 

3075 
7087 

9462 
3477 
7487 

9864 
3878 
7888 

♦0265 

4279 
8289 

*o667 
4680 
8690 

*i87i 
5884 
9892 

1084 
1085 
1080 

0350293 
4297 
8298 

0693 
4698 
8698 

1094 
5098 
9098 

1495 
5498 
9498 

1895 
5898 
9S98 

2296 

6298 

*0297 

2696 

6698 

*o697 

3096 

7098 

*i097 

3497 

7498 

♦1496 

3897 

7898 

*i896 

1087 
1088 
1089 

1090 

1 09 1 
1092 
1093 

036  2295 

6289 

0370279 

2695 
6688 
0678 

3094 
7087 
1076 

3494 
7486 

1475 

3893 
7885 
1874 

4293 
8284 
2272 

4692 
8683 
2671 

5091 
9082 
3070 

5491 
9481 
3468 

5890 
9880 
3867 

4265 

4663 

5062 

5460 

5858 

6257 

6655 

*o635 
4612 

8585 

7^3 

*io33 
5009 
8982 

7451 
*i43i 

5407 
■9379 

7849 

8248 

038  2226 

6202 

8646 
2624 
6599 

9944 
3022 
6996 

9442 
3419 
7393 

9839 
3817 
7791 

*o237 
4214 
8188 

♦1829 
5804 
9776 

1094 

1095 
1096 

0390173 
4141 
8106 

0570 
4538 
8502 

0967 

4934 
8898 

1364 
5331 
9294 

1761 

5727 
9690 

2158 

6124 

*oo86 

2554 

6520 

♦0482 

2951 

6917 

*o878 

3348 

7313 
*I274 

3745 

7709 

*i670 

1097 
1098 
1099 

MOO 

040  2066 
6023 
9977 

2462 

6419 

♦0372 

2858 

6814 

♦0767 

3254 
7210 

*Il62 

3650 
7605 

*i557 

4045 
8001 

♦1952 

4441 

8396 

*2347 

4837 
8791 

*2742 

5232 
9187 

*3i37 

5628 

9582 

*3532 

041  3927 

4322 

4716 

5"! 

•5506 

5900 

6295 

6690 

7084 

7479 

N. 

L.   0 

1 

2 

3 

4 

5 

6 

7     8 

9 

S." 

T." 

S."    T." 

2°  55'  =  10500"    4.( 

38539 

595 

3°o'=  10800"    4.68538    597 

2  56  =  10560 

539 

.595 

3  I  =  10860        537    598 

2  57  =  10620 

538 

596 

32=  10920        537    598 

2  58  =  10680 

538 

596 

3   3=  10980        537    599 

2  59  =  10740 

538 

597 

34=1 1040        537    599 

24 

3° 

/ 

M. 

S'.  T'. 

Sec.  1  S".  T".  ll 

6.46   1 

4.68   II 

0 

I 

180 

353 

412 

10800 

538 

597 

181 

353 

413 

10860 

537 

598 

2 

182 

352 

413 

10920 

537 

598 

3 

183 

352 

414 

10980 

537 

599 

4 

184 

352 

414 

1 1 040 

537 

599 

5 

185 

352 

41S 

moo 

537 

599  1 

i  6 

186 

351 

415 

I II 60 

536 

600 

7 

187 

351 

415 

1 1 220 

536 

600 

8 

188 

35 1 

416 

1 1 280 

536 

601 

9 
II 

189 

351 

416 

1 1 340 

536 

601 

190 

350 

417 

1 1400  :  535 

602 

191 

350 

417 

1 1460  1  535 

602 

12 

192 

350 

418 

1 1520 

535 

603 

13 

193 

350 

418 

1 1580 

535 

603 

14 

194 

350 

419 

1 1 640 

534 

604 

IS 

195 

349 

419 

1 1 700 

534 

604 

16 

196 

349 

420 

1 1 760 

534 

605 

17 

197 

349 

420 

1 1 820 

534 

605 

18 

198 

349 

421 

1 1 880 

533 

606 

19 

20 

21 

199 

348 

421 

1 1940 

533 

606 

200 

348 

422 

12000 

533 

607 

201 

348 

422 

12060 

533 

607 

22 

202 

348 

423 

1 21 20 

532 

608 

23 

203 

347 

423 

12180 

532 

608 

24 

204 

347 

424 

12240 

532 

609 

25 

20s 

347 

424 

12300 

532 

bo9 

26 

206 

347 

425 

12360 

531 

610 

27 

207 

346 

42s 

12420 

531 

610 

28 

208 

346 

426 

12480 

531 

611 

29 
30 

31 

209 

34t) 

426 

12540 

531 

611 

210 

346 

427 

12600 

530 

612 

211 

345 

427 

12660 

530 

612 

32 

212 

345 

428 

12720 

530 

613 

33 

213 

345 

428 

12780 

530 

613 

34 

214 

345  429 

12840 

529 

614 

35 

215 

344  1  429 

12900 

529 

614 

36 

216 

344  430 

12960 

529 

615 

37 

217 

344 

430 

13020 

529 

615 

^>i 

218 

344 

431 

13080 

528 

616 

39 
40 

41 

219 

343 

431 

13140 
13200 

528 
528 

616 
617 

220 

343 

432 

221 

343 

432 

13260 

528 

^'l 

42 

222 

342 

433 

13320 

527 

618 

43 

223 

342 

434 

13380 

527  1  618 

44 

224 

342 

434 

13440 

527  1  619 

45 

22s 

342 

435 

13500 

526  j  620 

46 

226 

341 

435 

13560 

526 

620 

47 

227 

341 

436 

13620 

526 

621 

48 

228 

341  i  436 

13680 

526 

621 

49 
50 

51 

229 
230 

340 
340 

437 

13740 

525 

622 

437 

13800 

525 

622 

231 

340  j  438 

13860 

525  i  623  II 

52 

232 

340  1  439 

13920 

525 

623 

53 

233 

339 

439 

13980 

524 

624 

54 

234 

339 

440 

14040 

524 

625 

55 

235 

339 

440 

14100 

524  i  625  II 

56 

236 

33^ 

441 

141 60 

523 

626 

57 

237 

338 

441 

14220 

523 

626 

S8 

238 

33ii 

442 

14280 

523 

627 

59 
60 

239 
240 

338 

337 

443 
443- 

14340 

522 

628 

14400 

522  1  628  II 

'   M.  1  S'.  T'.  1 

Sec.  1  S".  T".   1 

1 

6.46   1 

4.68 

0 

240 

337 

443 

14400 

522  !  628 

241 

337 

444 

14460 

522 

629 

2 

242 

337 

444 

14520 

522 

629 

3 

243 

336 

445 

14580 

521 

630 

4 

244 

336 

446 

14640 

521  1  631 1 

5 

245 

336 

446 

14700  521  1  631  1 

6 

246 

33(>    447 

14760 

520 

632 

7 

247 

335  447 

14820 

520 

632 

8 

248  1  335  :  448 

14880 

520 

633 

9 
10 

II 

249  335  i  449 

14940 

520 

634 

250  i  334  1  449 

251  i  334  !  450 

15000 

519 
519 

634 
635 

15060 

12 

252  1  334  1  450 

15120  1  519 

63s 

13 

253  333    451 

15180  j  518 

636 

H 

254 

333    452 

15240  i  518 

637 

15 

255 

333  1  452 

15300  518 

637 

16 

256 

332 

453 

15360  1  517 

638 

17 

257 

332 

454 

15420  1  517 

638 

18 

258 

332 

454 

15480  1  517 

639 

19 
20 

21 

259 
260 
261 

332 

455 

15540  1  516 

640 

331 

456 

15600  1  516 

640 

331 

456 

15660 

516 

641 

22 

262 

331 

457 

15720 

515 

642 

23 

263 

330 

457 

15780 

515 

642 

24 

264 

330 

458 

15840 

515 

643 

25 

26s 

330 

459 

15900 

514 

644 

26 

266 

329 

459 

15960 

5H 

644 

27 

267 

329 

460 

16020 

514 

645 

28 

268 

329 

461 

16080 

513 

646 

29 
30 

31 

269 

328 

461 

16140 

513 

646 

270 

328  1  462 

16200  1  513 

647 

271 

328 

463 

16260  1  512 

648 

32 

272 

327 

463 

16320  1  512 

648 

33 

273 

327 

464 

16380  1  512 

649 

34 

274 

327 

465 

16440  511 

650 

35 

275 

326 

465 

16500  511 

650 

36 

276 

326  !  466 

16560  1  511 

651 

37 

277 

326  i  467 

16620  1  510 

652 

38 

278 

325   467 

16680  510 

652 

39 
40 

41 

279 
280 

325 
325 

468 

16740  510 

653 

469 

16800  i  509 

654 

281 

324  '  469 

16860  1  509 

654 

42 

282 

324   470 

16920  509  1  655  1 

43 

283 

324  1  47» 

16980 

508 

656 

44 

284 

323  !  472 

17040 

508 

656 

45 

28s 

323  j  472 

17100 

508 

657 

46 

286 

323 

473 

17160 

507 

658 

47 

287  i  322 

474 

17220 

507 

659 

48 

288 

322  i  474 

17280 

507 

659 

49 
50 

51 

289 

321  !  475 

17340 

506 

b6o 

290 

321  j  476 

1740G  506  i  661 

291  1  321  1  477 

17460 

506  1  661 

52 

292  1  320  i  477 

17520 

505  j  662 

53 

293 

320  478 

17580 

505 

663 

54 

294 

320  !  479 

17640 

505 

664 

55 

29  s 

319 

479 

17700 

504 

bb4 

56 

296 

319 

480 

17760 

504 

665 

57 

297 

319 

481 

17820 

503 

666 

58 

298 

318 

482 

17880 

503 

666 

59 
60 

299 

318 

482 

17940 

503 

667 

300 

317 

483 

18000 

502 

668 

25 


r 

1 

II. 

1 

THE    LOGARITHMS 

• 

OF  THE 

TRIGONOMETRIC    FUNCTIONS 

FOR   EACH   MINUTE. 

Formulas  for  the  Use  of  the  Auxiliaries  5  and  T. 

1.    When  a  is  in  the  Hrst  five  degrees  of  the  quadrant : 

log  sin  a  =  log  a'  +  S.' 

log  a'   =  log  sin  a  +  cpl  SJ 

log  tan  a  =  log  a'  +  7'.' 

=  log  tan  a  +  cpl  7'.' 

log  cot  a  =  cpl  log  tan  a. 

=  cpl  log  cot  a  4-  cpl  7'.' 

log  sin  a  =  log  a"  +  S." 

log  a"  =  log  sin  a  +  cpl  5." 

log  tan  a  =  log  a"  +  TJ' 

=  log  tan  c  +  cpl  7"." 

log  cot  a  =  cpl  log  tan  a. 

=  cpl  log  cot  a  -1-  cpl  7." 

2.    When  a  is  in  the  last  five  degrees  of  the  quadrant : 

log  cos  a  =  log(90°  -  a)'  +  SJ 

log(90°  — a)'  =  log  cos  a  +  cpl  SJ 

log  cot  a  =  log(90°  -  a)'  +  TJ 

=  log  cot  a  +  cpl  y\' 

log  tan  a  =  cpl  log  cot  a. 

=  cpl  log  tan  a  +  cpl  T.' 

log  cos  a  3=  log(90°  -  a)"  +  6'." 

log(90°-a)"=  log  cos  a  -f  cpl  5." 

log  cota  =  log(90°  -  a)"  +  7V' 

=  log  cot  a  +  cpl  r." 

log  tan  a  =  cpl  log  cot  a. 

=  cpl  log  tan  a  +  cpl  7'." 

a  =  90° -(90° -a). 
/ 

26 

0° 

// 

/ 

L.  Sin. 

1    d. 

iCpl.  S'.  1  Cpl.  T'. 

L.  Tan.  jc.  d. 

L.  Cot. 

1  L.  Cos. 

o 

0 

646  373 

'^0101 

— 

— 

— 

30103 
17609 
12494 

— 

0.00000 

60 

59 

60 

3-53  627 

3.53627 

6.46  373 

3-53  627 

0.00000 

120 

2 

6.76476 

17609 
12494 

3-53627 

3-53627 

6.76476 

323  524 

0.00  000 

S8 

1 80 

3 

6.94  085 

353627 

3-53627 

6.94  085 

3-05915 

0.00  000 

57 

240 

4 

7.06579 

9691 

3-53  627 

3.53  627 

7.06579 

9691 
7918 
66q4 

2.93421 

0.00  000 

S6 

300 

5 

7.16  270 

3-53  627 

3-53627 

7.16  270 

2.83  730 

0.00  000 

55 

360 

6 

7.24  188 

6694 

3-53  627 

353627 

7.24  188 

2.75812 

^  0.00  000 

54 

420 

7 

7.30882 

5800 

3-53  627' 

3-53627 

7.30  882 

5800 

2.69  118 

1  0.00  000 

53 

480 

8 

7.36682 

3.53627 

353627 

7.36  682 

2.63318 

1  0.00  000 

52 

540 
600 

9 
10 

7.41  797 

4576 
4139 
3779 
3476 
3218 
2997 

3-53  627 

353627 

7.41  797 

4576 
4139 
3779 
3476 
3219 
2oa6 

2.58  203 

0.00  000 

51 
50 

49 

746373 

3-53627 

3-53627 

7-46  373 

2-53  627 

0.00000 

660 

750  5' 2 

3-53627 

3-53627 

7.50512 

2.49  488 

0.00000 

720 

12 

7.54291 

•3-53  627 

3-53627 

7-54  291 

2.45  709 

o.coooo 

48 

780 

13 

7-,57  767 

3-53627 

3.53  627 

7-57767 

2.42  233 

0.00  000 

47 

840 

14 

7.60  9S5 

3-53628 

3-53627 

7.60  986 

2.39014 

0.00  000 

46 

900 

15 

7.63  982 

2802 

3-53  628 

3-53627 

7.63  982 

2803 
2633 

2.36018 

o.co  000 

45 

960 

16 

7.66  784 

2633 

353628 

3-53  627 

7.66  785 

2.33215 

0.00  000 

44 

1020 

7 

7.69417 

2483 

3-53  628 

3-53  627 

7.69418 

2.30  582 

9-99  999 

43 

1080 

18 

7.71  900 

2348 
2227 
2119 

3-53  628 

3-53627 

7.71  900 

2348 
2228 
2119 

2.28  100 

9.99  999 

42 

1 140 

19 
20 

21 

7.74  248 

3-53  628 

353627 

7-74  248 

2.25  752 

9-99  999 

41 
40 

39 

1200 
1260 

7-76  475 

3.53628 

353627 

7.76476 

2.23  524 

9-99  999 

7-78594 

353628 

3-53  627 

7-78  595 

2.21  405 

9-99  999 

1320 

22 

7.80615 

3.53628 

3.53627 

7.80615 

2.19385 

9.99  999 

38 

1380 

23 

7-82  545 

1930 
1848 

353628 

3-53627 

7.82546 

1931 
1848 

2.17454 

9.99  999 

37 

1440 

24 

7-84  393 

3-53  628 

3-53627 

7-84  394 

2.15  606 

9  99  999 

36 

1500 

2S 

7.86  166 

353628 

3-53627 

7.86  167 

2.13833 

9-99  999 

35 

1560 

26 

7.87  870 

1639 

353628 

3-53627 

7.87871 

1639 

2.12  129 

9.99  999 

34 

1620 

27 

7.89  509 

1579 
1524 
1472 

1424 

3.53  628 

3-53626 

7.89510 

1579 
1524 
1473 
1424 

2.10490 

9  99  999 

V^ 

1680 

28 

7.91  088 

3-53  628 

3-53.626 

7.91  089 

2.08911 

9-99  999 

32 

1740 

29 

30 

31 

7.92  612 

3-53  628 

3-53626 

7.92613 

207387 

9.99  998 

31 
30 

29 

1800 

i860 

7.94084 

3-53  628 

3-53626 

7.94  086 

2.05  914 

9-99  998 

7-95  508 

3.53628 

3-53626 

7.95510 

2.04  490 

9-99  998 

1920 

32. 

7.96887 

1336 
1297 

3.53  628 

3-53  626 

7.96  889 

1336 
1297 

2.03  1 1 1 

9-99  998 

28 

1980 

33 

7.98  223 

3.53  628 

3-53  626 

7.98  225 

2.01  775 

9-99  998 

27 

2040 

34 

7.99  520 

3.53628 

3-53  626 

7.99  522 

2.00  478 

9-99  998 

26 

2100 

3S 

8.00  779 

353628 

3-53626 

8.00  781 

1.99  219 

9.99  998 

25 

2160 

36 

8.02  002 

1 190 
1158 

3-53  628 

353626 

8.02  004 

1190 

"59 
1 128 

1.97996 

9-99  998 

24 

2220 

37 

8.03  192 

3.53  628 

3-53  626 

803194 

1.96806 

9.99  997 

23 

2280 

3« 

8.04  350 

3-53  628 

3-53  626 

804  353 

1.95  647 

9.99  997 

22 

2340 
2400 

39 
40 

41 

S.05  478 

1 100 
1072 
1046 

353628 

3.53  626 

8.05  481 

IIOO 

1072 

1.94  5 19 

9-99  997 

21 
20 

19 

8.06  578 

3-53  628 

353625 

8.06581 

1.93419 

9-99  997 

2460 

8.07  650 

3-53  628 

3-53  625 

8.07  653 

1-92347 

9-99  997 

2520 

42 

8.08  696 

3-53  628 

353625 

8.08  700 

1.91  300 

9.99  997 

18 

2580 

43 

8.09718 

999 
976 

3-53  629 

353625 

8.09  722 

998 
976 

1.90  278 

9.99  997 

17 

2640 

44 

8.10717 

3-53  629 

3-53625 

8.10  720 

1.89280 

9.99  996 

16 

2700 

4S 

8. 1 1  693 

3.53  629 

3-53625 

8. 1 1  696 

1.88304 

9-99  996 

15 

2760 

46 

8.12647 

954 
934 

3.53629 

3-53625 

8.12651 

955 
934 

1-87349 

9  99  996 

14 

2820 

47 

8.13  581 

3-53  629 

3-53  625 

8.13585 

1.86  415 

9.99  996 

13 

2880 

48 

8.14495 

914 
896 
877 
860 

843 
812 

3-53  629 

3-53625 

8.14500 

H'5 

895 
878 

860 

843 
828 
812 

1.85  500 

9.99  996 

2940 

49 
50 

SI 

8.15391 

3-53629 

353624 

8.15395 

1.84605 

9.99  996 

11 
10 

9 

3000 
3060 

8.16268 

3-53  629 

3-53  624  I 

8.16273 

1.83727 

9.99  995 

8.17128 

3-53  629 

3.53  624 

8.17  133 

1.82867 

9-99  995 

3120 

S2 

8.17971 

3-53629 

3-53  624 

8.17976 

1.82024 

9-99  995 

8 

3180 

53 

8.18798 

3-53629 

3-53  624 

8.18804 

1.81  196 

9-99  995 

7 

3240 

S4 

8.19  610 

782 

769 

353629 

3-53  624 

8.19616 

797 
782 
769 
756 
742 
730 

1.80384 

9-99  995 

6 

3300 

ss 

8.20407 

3-53  629 

3-53  624 

8.20413 

1-79587 

9-99  994 

5 

3360 

56 

8.21  189 

3-53  629 

3  53  624 

8.21  195 

1.78  805 

9.99  994 

4 

3420 

57 

8.21  958 

TCC 

353629 

3-53623 

8.21  964 

1.78036 

9.99  994 

3 

3480 

58 

8.22713  ;r,  1 

3-53  629 

3-53623 

8.22  720 

1.77  280 

9.99  994 

2 

3540 

59 
60 

8.23456 

730 

3  53  630 

3-53623 

8.23  462 

1.76538 

9.99  994 

I 
0 

3600 

8.24 186 

3.53  630 

3-53  623 

8.24  192 

1.75  808 

9-99  993 

L.  Cos. 

d.       1 

1 

L.  Cot. 

c.  d. 

L.  Tan. 

L.  Sin. 

/ 

89' 


27 


I  L.  Sin. 


d.  I  Cpi.  s^  i  cpl.T^ 


L.  Tan.  c  d.  L.  Cot. 


L.  Cos. 


3600 

3660 
3720 
3780 
3840 

3900 
3960 
4020 
4080 
4140 
4200- 
4260 
4320 
4380 
4440 
4500 
4560 
4620 
4680 
4740 
4800 
"486^ 
4920 
4980 
5040 
5100 
5160 
5220 
5280 
5340 
5400 
5460 
5520 
5580 
5640 
5700 
5760 
5820 
5880 
5940 
6000 
6060 
6120 
6180 
6240 
6300 
6360 
6420 
6480 
6540 
6600 

~66fe 
6720 
6780 
6840 
6900 
6960 
7020 
7080 
7140 


7200 


o 

I 

2 

3 

4 
5 
6 

7 
8 

9 

iO 

II 
12 
13 
H 
»5 
16 

17 

18 

19 
20 

21 

22 
23 
24 

25 
26 

27 
28 

29 

30 

31 
32 

33 
34 
35 
36 

37 
38 
39 
40 

41 
42 
43 
44 
45 
46 

47 
48 

49 

50 

51 

52 
53 

54 
55 
56 

57 
58 
59 
60 


8.24  186 

8.24  903 

8.25  609 

8.26  304 
8.26  988 
8.27661 
8.28324 

8.28977 

8.29  621 

8.30  25^ 
8.30  879 


8.31  495 

8.32  103 

8.32  702 

8.33  292 
8.33875 

8.34  450 
8.35018 

8.35  578 

8.36  131 


8.36  678 
8.37217 

8.37  750 

8.38  276 

8.38  796 
8.39310 
8.39818 

8.40  320 
8.40816 

8.41  307 


8.41  792 


8.42  272 

8.42  746 

8.43  216 

8.43  680 

8  44  139 

8.44  594 

8.45  044 
8.45  489 
8.45  930 


.46  306 


8.46  799 

8.47  226 

8.47  650 

8.48  069 
8.48  485 

8.48  896 

8  49  304 

8.49  708 

8.50  108 


■50  504 


8.50  897 

8.51  287 
8.51673 
852055 
8.52434 
8.52810 
8.53  183 
8-53  552 
8.53919 


■54  282 


717 
706 
695 
684 

673 
663 

653 
644 
634 
624 
616 
608 
599 
590 
583 
575 
568 
560 
553 
547 
539 

533 
526 
520 

5H 
508 
502 
496 
491 

485 
480 

474 
470 
464 
459 
455 
450 

445 
441 

436 
433 
427 
424 
419 
416 
411 
408 
404 
400 
396 
393 
390 
386 

382 
379 
376 
373 
369 
367 
363 


3:53  ^30 
3-53  630 
3-53  630 
3-53  630 
3.53  630 
3-53  630 
3-53  630 
3-53  630 
3-53  630 
3-53  630 


3.53  630 


3-53  630 
353631 
3-53631 
353631 
353631 
353631 
353631 
3-5363' 
3-53631 


3.53631 


353631 
3-53  632 
3-53  632 
3-53  632 
353632 
3-53632 

3-53  632 
353632 
353632 


3-53  632 
3-53  632 
3-53  633 
3-53  633 
3-53  633 
353633 
3-53633 
3-53  633 
3-53633 
3-53  633 


3-53  634 


3-53  634 
3-53  634 
3-53  634 
3-53  634 
3.53  634 
3-53  634 
3-53  634 
3-53  635 
3-53  635 
3-53  635 
3-53  635 
3-53  635 
3-53635 
3-53635 
3-53  635 
3.53  636 
3-53  636 
3  53  636 
3_5  3  6^6 
3-53636 


3i53  623 
353623 
3-53623 
3-53  623 
3.53  622 
353622 
353622 
353622 
353622 
353622 


3-53621 


3-53621 
3-53621 
3-53621 
353621 
3.53  620 
3.53  620 
3.53  620 
3-53  620 
3-53  620 


3-53620 


3-53619 
3-53619 
3-53619 
3-53619 
3-53619 
3-53618 

3-53618 
3-53618 
3-53618 


3-53617 
3-53617 
3-53617 
3-53617 

3-53617 
3-53616 
353616 


3-5: 


616 


3-53616 
3-53615 


3-53615 


3-53615 
3-53615 
3-53614 
3-53614 
3-53614 
3-53614 
3-53613 
3-53613 
3-53613 


3-53613 
3-53612 
3-53612 
3-53612 

3-53611 
3-53611 
353611 
3-53611 
3-53610 
3-53  610 


•53610 


8.24  192 
8.24910 
8.25616 
8.26312 

8.26  996 

8.27  669 

8.28  332 

8.28  986 

8.29  629 

8.30  263 
8:30888 

8.31  505 
8.32112 

8.32  7" 

8.33  302 
8.33  886 
8-34461 
8.35  029 

8.35  590 

8.36  143 


8.36689 


8.37  229 

8.37  762 

8.38  289 

8.38  809 
8-39  323 

8.39  832 

8.40  334 

8.40  830 

8.41  321 


8.41  807 


8.42  287 

8.42  762 

8.43  232 

8.43  696 

8.44  156 

8.44  61 1 

8.45  061 

8.45  507 
8.45  948 


8.46  385 
8.46817 

8.47  245 

8.47  669 

8.48  089 

8.48  505 
8.48917 

8.49  325 

8.49  729 

8.50  130 


8-50527 

8.50  920 

8.51  310 
8.51  696 
8.52079 
8.52459 
8.52831 
8.53  208 
8.53578 

8.53  945 

8.54  308 


718 
706 
696 
684 

673 
663 

654 
643 
634 
625 

617 
607 

599 
591 
584 
575 
568 

561 
553 
546 
540 
533 
527 
520 

514 
509 
502 
496 
491 
486 
480 

475 
470 

464 
460 
455 
450 
446 
441 

437 
432 
428 
424 
420 
416 
412 
408 
404 
401 

397 
393 
390 
386 
383 
.380 
376 
373 
370 
367 
363 


75808 

75090 

74384 
73688 

73004 
72331 
71  668 

71  014 
70371 
69737 


9-99  993 


69  112 

68T95 
67888 
67289 
66698 
66  1 14 
65539 
64971 
64  410 
63857 


9-99  993 
9.99  993 
9  99  993 
9.99  992 
9  99  992 
9  99  992 
9.99  992 
9-99  992 
9i99  99i 
9.-99  99  y 
9.99991 
9.99  990 
9.99  990 
9.99  990 
9.99  990 
9.99  989 
9.99  989 
9  99  989 
9-99  989 


63311 


62771 
6:2238 
61  711 
61  191 
60  677 
60  168 
59  666 
59170 
58679 


9.99-988 
9.99  988 
9-99  988 
9-99  987 

9  99  987 
9.99  987 
9-99  986 
999986 
9.99  986 
9-99  985 


58193 


57713 
57238 
56768 

56304 

55844 
55389 
54  939 
54  493 
54052 


9-99  985 
9-99  985 
9-99  984 
9.99  984 

9.99  984 
9.99  983 
9-99  983 
9.99  983 
9.99  982 
9.99  982 


53615 


9.99  982 


53183 
52755 
52331 
51  911 
5M95 
51083 

50675 
50271 
49870 


9.99  981 
9.99  981 
9.99981 

9-99  980 
9.99  980 
9.99979 
9.99979 
9.99  979 
9-99978 


49  473 
49080 
48  690 
48304 
47921 
47541 
47  165 
46  792 
46  422 
46055 


9.99  978 
9.99977 
9.99977 
9.99977 

9-99  976 
9-99  976 
9-99  975 

9-99  97l 
9.99  974 

9-99  974 


45692 


9-99  974 


L.  Cos. 


L.  Cot.  |c.  d.  I  L.  Tan. 


L.  Sin. 


QQ° 


28 

2^ 

n 

/ 

L.  Sin.    j    d. 

LCpl.  S'. 

Cpl.  T'. 

1  L.Tan. 

c.  d.  1   L 

..  Cot. 

L.  Cos. 

7200 
7260 

0 

I 

8.54  282 

360 

357 
355 
351 

1  3-53  636 

3.53610 

8.54  308 

361     '• 

45692 

9-99  974 

60 

S9 

8.54  642 

3.53  636 

3.53  609 

8.54  669 

301 

45331 

9-99  973 

7320 

2 

8.54  999 

3-53  637 

3-53  609 

8.55  027    :^^r  i  I 

44  973 

9.99  973 

58 

•  7380 

3 

8-55  354 

3-53  637 

3.53609  i,  8.55382 

000         I 

44  618 

9.99972 

57 

7440 

4 

8-55  705 

3.53637 

3.53609     8.55734 

352         J 

44266 

9.99972 

S6 

7500 

S 

8.56054 

349 
346 

3.53  637 

3-53  608  1 

8.56083 

349     , 
346     , 

43917 

9.99971 

55 

7560 

6 

8.56  400 

3-53  637 

3-53  608 

8.56  429 

43571 

9.99971 

54 

7620 

7 

8.56  743 

343 

3.53637 

3-53  608 

8.56773 

344  1 

T   ,1    I        i 

43227 

9.99  970 

53 

7680 

8 

8.57084 

341 

3-53  637 

3-53  607  1 

8.57 114 

341       1        T 

338 1 ; 

42886 

9-99  970 

52 

7740 
7800 
7860 

9 
10 

II 

8.57421 

337 
336 
332 

3-53  638 

3-53  607  1 

18.57  452 
8.57  788 

42548 

9.99  969 

51 
50 

49 

8-57  757 

3-53  638 

3.53  607 

42  212 

9.99  969 

8.58089 

3.53  638 

3-53  606 

8.58  121 

ZZZ  1  J 

41  879 

9.99  968 

7920 

12 

8.58419 

ZIP 
328 

325 

3.53  638 

3-53  606 

8.58451 

330  \  , 

328  i 
326  ^ 

41  549 

9.99  968 

48 

7980 

13 

8.58  747 

3.53  638 

3.53  606 

8.58  779 

41  221 

9.99  967 

47 

8040 

14 

8.59072 

3-53  638 

3-53  605 

8.59  105 

40895 

9.99  967 

46 

8100 

15 

8.59  391 

323 
320 

318 

316 

3-53  639 

3-53  605 

8.59428 

323    I 

40572 

9.99  967 

45 

8160 

16 

8.59715 

3-53  639 

3-53605  !|  8.59749 

321     J 

40251 

9.99  966 

44 

8220 

17 

8.60  033 

3-53  639 

3.53  604  \ 

8.60  068 

319    , 

316  ; 

39932 

9.99  966 

43 

8280 

18 

8.60  349 

3-53  639 

3.53  604  i 

8.60  384 

39616 

9.99  965 

42 

J340 

8400 

~876o 

19 
20 

21 

8.60  662 

zn 

3" 

309 

3.53  639 

3-53  604  I 

8.60  698 

3H     I 

39302 

9.99  964 

41 
40 

39 

8.60973 

3.53  639 

3-53  603 

8.61  009 

31 '     J 

38991 

9.99  964 

8.61  282 

3-53  640 

3-53  603  i 

8.61  319 

310 

35681 

9.99  963 

8520 

22 

8.61  589  j  ^^/ 

3-53  640 

3.53  603  i 

8.61  626 

307     , 

38  374 

9-99  963 

38 

8580 

23 

8.61  894  !  -^^s 

3-53  640 

3-53  602  ; 

8  61  931 

305     I 

38069 

9.99  962 

Zl 

8640 

24 

8.62 196 !  302 

3-53  640 

3.53  602   j  8.62  234 

303     J 

37  766 

9  99  962 

36 

8700 

25 

8.62  497 

298 
296 

3-53  640 

3.53602  i 

8.62  535 

301     I 

299  1  I 

.37465 

9.99  961 

35 

8760 

26 

8.62  795 

3-53  640 

3-53  601  i 

8.62  834 

3716b 

9.99961 

34 

8820 

27 

8.63  091 

3-53  641 

3.53601  i!.  8.63  131 

297  1  I 

_    !    * 

36869 

9.99  960 

33 

8880 

28 

8.63  385 

294 

3.53641 

3.53601    i  8.63426  \^yji\\ 

36574 

9.99  960 

32 

8940 
9000 
9060 

29 
30 

31 

8.63  678 

293 

3-53641 

3.536001!  8.63718 

^9^  1  I 

36  282 

9-99  959 

31 
30 

29 

8.63  968 

290 

288 

287 
284 

283 
281 

3-53  641 

353600  j 

,8.64009 

291 

35991 

9-99  959 

8.64  256 

3-53641 

3.53599 

8.64  298 

287  ; 
285  I 
284 
281  , 
280  I 
278  , 
276  I 

35702 

9.99  958 

9120 

32 

8.64  543 

3.53  642 

3-53  599 

8.64  585 

35415 

9-99  958 

28 

9180 

ZZ 

8.64  827 

3-53  642 

3.53  599 

1  8.64  870 

35  130 

9.99  957 

27 

9240 

34 

8.65  no 

3-53  642 

3-53  598  i 

8.65  154 

34846 

999956 

26 

9300 

35 

8.65  391 

3-53  642 

3.53598  {j  8.65435 

34565 

9.99  956 

25 

9360 

36 

8.65  670 

^yy 

3-53  642 

3.53598   j  8.65  715 

34285 

9-99  955 

24 

9420 

37 

8.65  947 

2y; 
276 

3.53  642 

3-53  597  il  8.65993 

34007 

9.99  955 

23 

9480 

38 

8.66  223 

3-53643 

3-53  597   1  8.66  269 

33731 

9-99  954 

22 

9540 
9600 
9660 

39 
40 

41 

8.66497 

2;4 
272 
270 
269 

3-53  643 

3.53  596   i  8.66  543 

274  I 

33  457 

9-99  954 

21 
20 

19 

8.66  709 

3-53  643 

353596  ':  8.66816 

273  I 

33  184 

9.99  953 

8.67  039 

3-53  643 

3.53596  \\  8.67087 

271  ^ 

269  I 
268  , 
266 
264  , 
263  , 

32913 

9.99952 

9720 

42 

8.67  308 

3-53  643 

3-53  595   '  8.67356 

32644 

9.99952 

18 

9780 

43 

8-67  575     ^^7 
8.67841     ^f 
8.68  104  i  263 
8.68  367  1  263 

3-53  644 

3-53  595  il  8.67624 

32376 

9-99951 

17 

9840 

44 

3-53  644 

3-53  594  ;|  8.67890 

32  no 

9.99951 

16 

9900 

45 

3.53  644 

3.53594:!  8.68154 

31  846 

9-99  950 

15 

9960 

46 

3.53  644 

3-53  594  : 

8.68417 

31583 

9  99  949 

14 

10020 

47 

8.68627     260 

3.53  644 

3-53  593  1 

8.68  678 

261 

260  I 
258  , 

31322 

9  99  949 

13 

looSo 

48 

8.68  886     259 

3-53  645 

3.53593  ji  8.68938 

31  062 

9.99  948 

12 

10140 
10200 
10260 

49 
50 

51 

8.69  144 

256 
254 

3-53  645 

3.53592  1!  8.69196 

30804 

9.99  948 

II 
10 

9 

8.69  400 

3.53  645 

3-53592  !|  8.69453 

257  J 

30  547  r  9-99  947 

8.09  054 

3-53  645 

3.53592  !l  8.69708 

255  J 

30292 

9.99946 

10320 

52 

8.69907. 

253 

3-53  646 

3.53591     8.69962 

254  I 

30038 

9-99  946 

8 

10380 

53 

8.70159 

252 

3.53  646 

3-53  591    !  8.70214 

252  1 

29786 

9-99  945 

7 

10440 

54 

8.70409 

250 

3-53  646 

3-53590  1'  8.70465 

1^0  ' 

29535 

9.99  944 

6 

10500 

55 

8.70658 

249 

3-53  646 

3.53590  !i  8.70714 

249  I 
248  I 

2(19286 

9-99  944 

5 

10560 

56 

8.70  905  1  247 

3.53  646 

3-53589  ,;  8.70962 

29038 

9-99  943 

4 

10620 

57 

8.71  151 

24b 

3-53  647 

3.53  589!!  8.71208 

246  J 

28792 

9-99  942 

3 

to68o 

S8 

8-71  395 

244 

3-53  647 

.3-53589  li  8.71453 

245  I 

28547 

9-99  942 

2 

10740 
10800 

59 
60 

8.71  638 

243 

3  53  647 

3.53  588 

'  8.71  697 
j  8.71  940 

244  I 

28. 303 

9-99  941 

I 
0 

8.71  880 

.4. 

3.53  647 

3-53  588 

243  J 

28060 

9.99  940 

L.  Cos. 

d.     1 

1!    L.  Cot. 

c.  d.  1  L 

.Tan. 

L.  Sin. 

t 

87° 


29 


L.  Sin. 


L.  Tan. 


c.d. 


L.  Cot.   L.  Cos. 


P.P. 


O 

I 

2 

3 
4' 
5 
6 

7 
8 

9 

10 

II 

12 
13 
H 

«5 
16 

17 

18 

19 
20 

21 

22 
23 
24 
25 
26 

27 
28 
29 

30 

31 
32 
33 
34 

35 
36 

38 
39 
40 

41 

42 

43 

44 

45 
46 

47 
48 
49 
50 

51 

52 
53 
54 
55 
56 

57 
58 
59 
60 


8.71  880 

8.72  120 

8.72  359 
872  597 

8.72834 
8.73009 

8.73  303 
8-73  535 
8.73  7<^7 
8-73  997 


8-74  220 

8.74  454 
8.74680 

8.74  906 

8-75  130 
8-75  353 
8-75  575 

8.75  795 
8.75015 

876  234 


8.76451 


8.76667 

8.76  883 

8.77  097 
8.77310 
8.77522 
8.77  733 

877943 
8.78152 
8.78360 
8.78568 


8.78  774 

8.78  979 

8.79  183 
8.79  386 
8.79  588 

8.79  789 
8.79990 

8.80  189 

8.80  T.^'i 


.80  585 


8.80  782 

8.80  978 

8.81  173 
8.81  367 
8.81  560 
8.81  752 

8.81  944 

8.82  134 
8.82  324 


8.8251, 


8.82  701 

8.82  888 
8.83075 

8.83  261 
8.83446 
8.83  630 
8.83813 

8.83  996 

8.84  177 


8.84358 


240 

239 
238 
237 
235 
234 
232 
232 
230 
229 
228 
226 
226 
224 
223 
222 
220 
220 
219 
217 
216 
216 
214 

213 
212 
211 
210 
209 
208 
208 
206 
205 
204 
203 
202 
201 
201 
199 
199 
197 
197 
196 

195 
194 

193 

192 

192 
190 
190 
189 
188 
187 
187 
186 

185 
184 

183 

183 
181 
181 


8.71  940 


8.72  181 
8.72420 
8.72659 

8.72  896 
8  73  «32 

8.73  366 
8.73  600 
873  832 
8.74063 


8.74  292 


8.74521 

8.74  748 

8.74  974 

875  199 

875  423 

8.75  645 

8.75  867 

8.76087 

8.76  306 

876525 

8.76  742 
8.76958 
877173 

877  387 

8.77  600 

8.77  8u 
8.78022 

8.78  232 
8.78441 


8.78649 


8.78855 
8.79061 
8.79  266 
8.79470 

8.79  673 
8.79875 
8.80076 

8.80  277 
8.80  476 


5.80  674 


8.80  872 

8.81  068 
8,81  264 
8.81  459 
8.81  653 

8.81  846 

8.82  038 
8.82  230 
8:82420 


8.82610 


8.82  799 
8.82987 

8.83  175 
8.83  361 
8.83  547 

8.83  732 
8.83916 

8.84  100 
8.84  282 


8.84  464 


241 

239 
239 
237 
236 
234 
234 
232 
231 
229 
229 
227 
226 
225 
224 
222 
222 
220 
219 
219 
217 
216 

215 
214 
213 
211 
211 
210 
209 
208 
206 
206 
205 
204 
203 
202 
201 
201 
199 
198 
198 
196 
196 

»95 
194 

193 
192 
192 
190 
190 
189 
188 
188 
186 
186 
185 
184 
184 
182 
182 


28  060 
^819 
27  580 
27  .341 
27  104 
,26  868 
26  634 
,26  400 
,26  168 
:15_937 
25  708 

25  479 
25  252 
,25  026 
24  801 
24  577 
24355 
24  ^M 
23913 
,23  694 


9.99  940 

9.99  940 

9-99  939 
9.99  938 

9-99  938 
9-99  937 
9-99  936 
9.99  936 
9-99  935 
9-99  934 
9-99  9J4 

9-99  933 
9.99  932 
9.99  932 

999931 
9.99  930 
9.99  ^29 
9.99  929 
9.99  928 
9.99  927 


23475 


9.99  926 


23258 
23042 
22  827 
22613 
22  400 
22  189 
21978 
21  768 
21  559 


9.99  926 

9-99  925 
9.99  924 

9.99  923 
9.99923 
9.99  922 
9.99921 
9.99  920 
9.99  920 


21  351 


9.99919 


,21  145 
20  939 
20  734 
20530 
20327 
20  125 
,19924 

19723 
.19524 


9.99918 
9.99917 
9.99917 
9.99916 
9.99915 
9.99914 

9-99913 
999913 
9.99912 


19326 


9.9991 


19  128 
18932 
18736 

18  541 

18347 
18  154 

,17  962 
,17770 
.17580 


9.99910 

9.99  909 
9.99  909 

9.99  908 
9.99  907 
9  99  906 
9.99  905 
9.99  904 
9.99  904 


17390 


9.99  903 


17  201 
17013 
16825 
16639 

16453 
16268 

16084 
15900 
15  718 


9.99  902 
9.99  901 
9  99  900 
9.99  899 
9.99  898 
9.99  898 
9.99  897 
9.99  896 
9-99.895 


15536 


9.99  894 


60 

59 

58 
57 
56 
55 
54 

53 
52 
51 
50 

49 
48 
47 
46 
45 
44 

43 
42 
41 
40 

39 
38 
37 
36 
35 
34 

2,2, 
32 
31 
30 

29 
28 
27 
26 

25 
24 

23 
22 
21 
20 

19 
18 

17 
16 
15 
14 

13 
12 
II 
10 

9 
8 

7 
6 

5 
4 

3 

2 

I 

O 


241    239    237 


\e.i 


24.1  23.9  23.7 

48.2  47.8  47.4 

72.3  71.7  71.1 

96.4  95.6  94.8 

120.5  119. 5  118. 5 

144.6  143.4  142.2 

168.7  '67.3  165.9  165.2  163.8 

192.8  191. 2  189.6  188.8  187.2 

216.9  215.1  213.3  212.4  210.6 


236    234 

23.6 

47-2 

70.8     70.2 

94.4  93.6 
1 18.0  117.0 
141.6  140.4 


232     231 


229 

22.9 
45- 


23.2     23.1 

46.4     46.2 

69.6     69.3 

92.8     92.4 
116.0  115.5 
139.2  138.6  137.4 
[62.4  161.7  160.3 
185.6  184.8  183.2 


68.7 

91.6 

4.5 


227 

22.7 
45-4 
68.1 
90.8 

;>3-5 


226 

22.6 
45.2 
67.8 
90.4 
3-0 


136.2  135.6 
[58.9  158.2 
[81.6  180.8 


208.8  207.9  206.1  204.3  203.4 


224 

22.4 
44.8 
67  2 
89.6 
112.0 
134-4 


222 

22.2 
44-4 

66.6 


220      219     217 


22.0 
44.0 
66.0 


0.0 
32.0 


[II. o 

:33-2 

56.8  155.4  154.0 

79.2  177.6  176.0 

201.6  199.8  198.0 


21.9 
43-8 
65.7 
87.6 
109.5 
131-4 
153-3 
175-2 
197.1 


21.7 
43-4 

65.1 
86.8 
108.5 
130.2 
151.9 
173-6 
195-3 


214      213       211     209 


216 

21.6     21.4     21.3 

43.2     42.8     42.6 

64.8     64.2     63.9 

86.4  85.6  85.2 
108.0  107.0  106.5 
129.6  128.4  127.8 
151.2  149.8  149.1 
172.8  171. 2  170.4  168.8 
194.4  192.6  191. 7   189.9 


42.2 
63-3 
84.4 
05-5 
26.6 
47-7 


20.9 
41.8 
62.7 
83.6 
104.5 
125-4 
146.3 
167.2 


208    206 

20.8  20. 6 
41.6  41.2 
62.4  61.8 
83.2  82.4 
104.0  103.0 
124.8  123.6 
145.6  144.2 
166.4  164.8 
187.2  185.4 


203      201      199 


20.3  20.1 

40.6  40.2 

60.9  60.3 

81.2  80.4 

101. S    TOO. 5 

121.8  120.6 
142  I  140.7 

162.4  160.8 

182.7  180.9 


198 

196 

I 

19.8 

19.6 

2 

39-6 

39-2 

3 

59-4 

58.8 

4 

79-2 

78.4 

e 

99.0 

q8.o 

6 

118.8 

117.6 

7 

138.6 

137.2 

8 

158.4 

156-8 

9 

178.2 

176.4 

18.8 

37-6 
56-4 
75.2 
94.0 

112.8 

t3i-6 
[50.4 
169.2 


186 

18.6 

37-2 

55-8 

74-4 

93.0 
[11.6 
[30.2  128.8 
[48.8  147.2 
[67.4  165.6 


194 

19.4 
38.8 
58.2 
77-6 
97.0 
116.4 
135-8 
155-2 
174.6 

184 

18.4 
36.8 
55  2 
736 
92.0 
no. 4 


19.2 
38.4 
57-6 
76.8 
96.0 
115-2 
134-4 
153-6 
172.8 


36-4 
54-6 
72.8 
91.0 
109.2 
127.4 
145.6 
1638 


19.9 
39-8 
59-7 
79.6 

99-5 
119.4 

139-3 
159.2 
179.1 


19.0 
38.0 
57.0 
76.0 
950 
1 14.0 
133.0 
152.0 
171. o 


36.2 
54-3 
72.4 
90-5 
108.6 
126.7 
144-8 
162.9 


L.  Cos. 


d. 


L.  Cot. 


c.d, 


L.  Tan.      L.  Sin 


p.  p. 


30 


I  L.  Sin.  I  d. 


L.  Tan.  c.d.  L.  Cot. 


L.  Cos. 


P.  P. 


O 

I 

2 

3 

4 

5 
6 

7 
8 

9 

10 

II 

12 

13 

•5 
i6 

17 

18 

19 
20 

21 

22 
23 
24 

25 
26 

27 
28 
29 

30 

31 
32 
33 

34 
35 
36 

37 
3« 
39 
40 

41 
42 
43 
44 
45 
46 

47 
.48 

49 
50 

51 

52 

54 

55 
56 

57 
5« 
59 
60 


^•^4  35^ 


«-«4  539 
8.84718 

8.84  897 
8.85075 

8.85  252 
8.85.129 
8.85  605 
8.85  780 
^■^5  955 


8.80  126 


8.86301 
8.86474 
8.86  645 
8.86816 

8.86  987 

8.87  156 
8.87  325 
8.87  494 
8.87661 


8.87829 
8.87995 
8.88  161 
8.88  326 
8.88  490 
8.88  654 
8.88817 

8.88  980 

8.89  142 
^  89  304 


8.89  464 


8.89  625 
8.89  784 

8.89  943 

8.90  102 
8.90  260 
8.90417 
8.90574 
8.90  730 
8.90  885 


8.91  040 


b.91  195 
8.9 1  349 
8.91  502 
8.91  655 
8.91  807 

8.91  959 

8.92  no 
8.92  261 
8  92  41  r 


8.92  .s6i 


8.92  710 

8.92  859 

8.93  007 

§93T5Tr 
8  93  301 
8.93  448 

8  93  594 
8  93  740 

S  93  m 
8  94030 


8.84464 


8.84  646 

8.84  826 

8.85  006 
8.85  185 
8.85  363 
8.Ss  540 
8.85  717 
8.85  893 
8.86069 


8.80  243 


6.86  417 
8.86591 

8.86  763 
8.86935 

8.87  106 
8.87  277 

8.87447 
8.87616 

8.87785 


^■^7  953 


8.88  120 
8.88  287 
8.88453 
8.88618 
8.8878} 

8.88  948 

8.89  III 
8.89  274 
8.89  437 


8.89  598 


8.89  760 

8.89  920 

8.90  080 
8.90  240 
8.90  399 
8.90  557 
8.90715 

8.90  872 

8.91  029 


8.91  165 


8.91  340 
8.91  495 

8.91  C5J 
891  803 

891  957 

8.92  no 

8.92  262 
8.92414 
8.92  565 


89^ 


8.92  866 

8.93  016 
8.93  165 

8-93  3' 3 
893462 
8.93  609 

8.93  756 

8-93  903 

8.94  049 


8.94  195 


5536 


9.99  894 


5  354 
5  174 
4  994 
4815 

4637 
4460 

4283 
4  107 
3931 


9.99  893 
9.99  892 
9.99  891 
9.99891 
9.99  890 
9.99  889 
9.99  888 
9.99  887 
9.99  880 


3  757 
3583 
3409 
3237 
3065 
2894 
2723 

2553 
2384 
2  215 


9-99  885 
9.99  884 
9.99  ms 
9.99  882 
9.99  881 
9.99  880 
9.99  879 
9.99  879 
9.99  878 
9-99  877 


2047 


9-99  876 


1  880 
I  713 
J  547 
1382 
"I  217 
I  052 
0889 
o  726 
0563 


9-99  875 
9.99  874 
9-99  873 
9.99  872 
9.99871 
9.99  870 
9.99  869 
9.99  868 
9-99  867 


0402 


9.99  866 


o  240 
0080 
,09  920 
.09  760 
.09  601 
•09  443 
.09  285 
.09  128 
.08  971 


9.99  865 
9  99  864 
9.99  863 
9.99  862 
9.99  861 
9.99  860 

9-99  859 
9.99  858 
9-99  857 


08815 


9-99  856 


08660 
oS  505 
08350 
08  197 
08043 
07  890 

07738 
07586 

07435 


07  284 


07  134 
,06  984 
06835 
06  687 
06  538 
06391 
06  244 
06097 
05951 


05  805 


9-99  855 
9.99  854 
9-99  853 
9.99  852 
9.99  851 
9  99  850 
9.99  84S 

9  99  847 
9.99  846 


9-99  845 


9.99  844 
9.99  843 
9  99  842 
9.99841 
9.99  840 
9-99  839 
9.99  838 

9-99  837 
9.99  836 


60 

59 

58 
57 
56 
55 
54 

53 
52 
51 
50 

49 
48 
47 
46 
45 
44 

43 
42 

41 

40 

39 
3^ 
37 
36 
35 
34 

33 
32 
31 
30 

29 
28 
27 
26 

25 
24 

23 
22 
21 
20 

19 
18 

17 
16 
15+T 


9-99  834 


182 

181 

ISO 

1 

18.2 

18.1 

18.0 

2 

36.4 

36.2 

36.0 

3 

54-b 

54-3 

540 

4 

72.8 

72.4 

72.0 

5 

91.0 

QO.s 

1,0.0 

b 

109.2 

108  6 

108.0 

7 

1274 

126.7 

126.0 

8 

145.6 

144.8 

144-0 

9 

163.8 

162.9 

162.0 

177 

176 

175 

I 

17.7 

176 

17-5 

2 

35  4 

35-2 

35-0 

3 

53-1 

52.8 

52.. s 

4 

70.8 

70.4 

70  0 

5 

88.  s 

88.0 

«7.S 

6 

106.2 

105.6 

105.0 

7 

123.9 

1232 

122.5 

b 

141.0 

140.8 

140.0 

9 

159-3 

158.4 

157.5 

179 

17.9 

35-3 
53-7 
71.6 
^9-5 
107.4 
125.3 
143-2 
161. 1 


17.4 
34.8 
52.2 
69^6 
87.0 
104.4 
121.8 
139.2 
is6.6 


178 

17.8 

35-6 

53-4 

71.2 

89.0 

106.8 

124.6 

142.4 

160.2 


17-3 
34-6 
5'-9 
69.2 
86.5 
103.8 
121. 1 
138.4 
155-7 


172      171      170 


17.2 
34-4 
51-6 
68.8 
86.0 
103.2 
120.4 
137-6 
1548 


17. 1 
34-2 
51-3 
68.4 
85-5 
102  6 
119.7 
136.8 


170 
34-0 
51.0 
68.0 
85  o 


169 

16.9 


168 

16.8 
33-8  33-6 
50.7  50.4 
67.6  67.2 
84.5  84.0 
02.0  101.4  100.8 
19.0  118.3  117.6 
36.0  135.2  134.4 
530  152.1   151.2 


167     166      165     164  163 

16.7  16.6  16.5  16  4  16.3 
33-4     33-2     33.0  32.8  32.6 

50.1  49.8     49.5  49.2  48.9 

66.8  66.4  66.0  65.6  65.2 
83.5     830    82.5  82.0  81.5 

100.2  99.6  99.0  98.4  97.8 
116.9  u6.2  115.5  114.8  114. 1 
133.6  132.8  132.0  131  2  130.4 

150.3  149.4  148.5  147.6  146.7 

162       161      160  159  158 

16.2  16.1  16.0  15.9  15.8 
32  4  32  2  32.0  31.8  31.6 
48  6  48.3  48  o  47.7  47.4 
64  8  64.4  64.0  63.6  63.2 
81.0  80.5  80.0  79.5  79.0 
97.2    96.6    96.0  95.4  94  8 

113. 4  1127  112.0  III. 3  ito.6 
129.6  128.8  128.0  127.2  126.4 
145.8  144.9  ^44-o  143-1  142.2 


157 

15.7 
31-4 
47-1 
62.8 
785 
94  2 
109.9 
125-6 
141-3 


156 

15-6 
31.2 
46.8 
62.4 
78.0 
936 
[09 


155     154     153 


15-5 

31.0 

46.5 

62.0 

77-5 

93  o 

0S.5  107 
124.8  124.0  123  2 
140.4  139.5  1386 


15-4 
30.8 
46.2 
61.6 
77.0 
92.4 


149 

149 
29.8 
44  7 
59-6 
74-5 
894 
106  4  105.7  105.0  104.3 
121  6  120.8  120.0  119. 2 
136.8  135.9  135-0  134. 1 


152 

151 

15.2 

I5-I 

30.4 

30.2 

45  b 

45-3 

bo.8 

bo.4 

76.0 

75-5 

91.2 

90.6 

150 
15.0 
30.0 
45-0 
60  o 
750 
90.0 


15-3 
30.6 

45-9 
61.2 

76.5 
91.8 
107.1 
122.4 
137-7 


148 

14.8 
29.6 
44-4 
59-2 
74.0 
88  8 
103.6 
118.4 
133  2 


I  L.  Cos. 


L.  Cot.  0.  d 


L.  Tan. 


L.  Sin. 


P.  P. 


fifi° 


31 


I  L.  Sin. 


d.  I  L.  Tan.  |c.d.  j  L.  Cot. 


L.  Cos. 


P.P. 


O 

I 

2 

3 
4 
5 
b 

7 
8 

9 

iO 

II 

12 

13 
14 

15 
16 

17 

18 

19 
20 

21 
22 
23 

24 

25 
26 

27 
28 
29 
30 

31 

32 
33 
34 
35 
36 

37 
3« 
39 
40 

41 

42 

43 
44 
45 
46 

47 
48 

49 

50 

5' 

52 
53 
54 
55 
56 

57 
5« 
59 
60 


8.94  030 


8.94  '74 
J^.94317 
8.94  461 

8.94  603 
8.94  746 

8.94  887 

8.95  029 

8.95  170 
8^5310 


8-95_4|o_ 

8.95  5^9 
8.95  728 

8.95  867 

8.96  005 
8.96  143 
8.96  280 
8.96417 

^•96553 
8.96  689 


8.96  825 


8.9b  960 
8.97  095 
8.97  229 

8.97  363 
8.97  49b 
8.97  629 
8.97  762 

8.97  894 

8.98  026 


8.98157 


8.98  288 
8.98419 
8.98  549 
8.98  679 

8.98  808 
8.98937 

8.99  066 
8.99  194 
8.99  322 


8.99  450 


8.99  577 
8.99  704 
8.99  830 

8.99956 
9.00  082 
9.00  207 
9.00  332 
9.00456 
9  00  581 


9.00  704 


9.00  828 
9.00951 

9.01  074 
901  196 
9.01  318 
9.01  440 
9  01  561 
9.01  682 
901  803 


9.01  923 


144 

H3 
144 

142 

143 
141 

142 
141 
140 
140 

139 

139 
139 
138 
138 
^37 
137 
136 
136 
136 
135 
135 
134 
134 
133 
133 
133 
132 
132 
131 
131 

131 

130 

130 

129 
129 
129 
128 
128 
128 
127 
127 
126 
126 
126 
125 
125 
124 

125 
123 

124 

123 
123 
122 
122 
122 
121 
121 
121 
120 


8.94  195 
8.94  340 
8.94  485 
8.94  030 

8.94  773 
8.94917 

8.95  000 
8.95  202 
8.95  344 
8.95  4«6 


8-95  ^27 


8.95  767 

8.95  908 

8.96  047 
8.96  187 
8.96  325 
8.96  464 
8.96  602 
8.96  739 
8.96  877 


8.97013 


8.97  150 
8.97  285 
8.97421 

8-97  556 
8.97  691 
8.97825 

8.97  959 

8.98  092 
8.98  225 


8.98  358 


8.98  490 
8.98  622 
8.98  753 

8.98  884 
8-99015 

8.99  145 
8.99  275 
8.99  405 
8.99  534 


8.99  662 


8.99  791 
8.99919 
9.00  046 

9.C0  174 
9.C0301 
9.00427 

9-00553 
9.C0  b79 
9.00  805 


9.00  930 


9.0  J  055 

9.01  179 
9.01  303 
9.01  427 
9  01  550 
9.01  673 

9.01  796 
9.01  918 
g. 02  040 


9.02  ib2 


M5 
•45 
145 
H3 
144 
M3 
142 
142 
142 
141 
140 
141 
139 
140 

138 
139 
138 
137 
^3^ 
136 

137 
135 
136 

135 
135 
134 
134 
133 
^33 
133 
132 
132 
131 
131 

131 

130 

130 
130 
129 
128 
129 
128 
127 
128 
127 
126 
126 
126 
126 
125 
125 
124 
124 
124 
123 
123 
123 
122 
122 
122 


1.05805 


1 .05  660 

105  5»? 
1.05  370 

1.05  227 
1.05083 
1 .04  940 
1 .04  798 
1.04  656 

104373 


9-99  834  60 


9.99  ii33 
9.99  832 
9.99831 
9.99  830 
9.99  829 
9.99  828 
9.99  827 
9.99  825 
9.99  824 


9-99  823 


1.04233 
1 .04  092 
1-03953 
1.03  813 
1-03673 
I;03  53^ 
1.03398 
1.03  261 
1.03  123 


9.99  822 
9.99  821 
9.99  820 
9.99819 
9.99817 
9.99  816 
9.99815 
9.99814 
9-99813 


1.02  987 


9.99  812 


1.02  850 
1.02  715 
1.02579 
1.02444 
1.02  309 
1-02  175 
1.02  041 
1. 01  908 
i.oi  775 


9.99  810 
9-99  809 
9.99  808 

9-99  807 
9.99  806 
9.99  804 
9.99  803 
9.99  802 
9.99  801 


I.OI  642 


9.99  8CO 


I.OI  510 
I.OI  378 
I.OI  247 
I.OI  116 
1.00985 

1.00855 

1. 00  725 
1. 00  595 
1 .00  466 


.00338 


1 .00  209 
1 .00  08 1 
0.99  954 
0.99  826 
0.99  699 
0.99  573 
0.99  447 
0.99321 
0.99  195 


0.99  070 


0.98  945 
0.98821 
0.98  697 

0.98  573 
0.98  450 
0.98  327 
0.98  204 
0.98  082 
0.97  960 


0.97  838 


L.  Cos.  I  d.  I  L.  Cot,  led.  L.  Tan.  |  L.  Sin.  | 


■99  798 
■99  797 
•99  796 

..99795 
9-99  793 
9.99  792 

9.99  791 
9.99  790 
9-99  788 


9-99  787 


,  -99  786 
9-99  78I 
999  783 
,-99782 
9.99781 
9.99  780 

9-99  778 
9.99  777 

9-99  776 


999  775 


9.99  773 
9-99  772 
9.99771 

9-99  769 
9.99  768 

9-99  767 

9-99  765 
9.99  764 

•99  763 


9.99  761 


59 

58 
57 
56 
55 
54 

53 
52 
51 
50 

49 
48 
47 
46 
45 
44 

43 
42 

41 

40 

39 
38 
37 
36 
35 
34 

33 
32 
31 
30 

29 
28 
27 
26 

25 
24 

23 
22 
21 

20 


147 

147 

294 
44.1 
588 
73  5 
88.2 
102.9 
117. 6 
132.3 


146 
146 
29  2 
438 
58.4 
73  o 
87.6 

102. 2 
I16.8 
I3I4 


I 

143 

14.2 

2 

28.6 

284 

S 

42.9 

426 

4 

57-2 

56.8 

t^ 

71-5 

71.0 

6 

85.8 

85.2 

7 

100. 1 

99.4 

8 

"4  4 

113. 6 

9 

128.7 

127.8 

13-9 
•27.8 
41.7 
55.6 
69.5 
834 
97-3 
III. 2 
125.1 

135 

13-5 
27.0 
40-5 
54-0 
67.5 


94.5 
108.0 


131 

I3-I 
26.2 

39-3 
52.4 
655 
78.6 
91  7 
104.8 
117.9 


127 
12.7 
25-4 
38.1 
50.8 
635 
76  2 
88.9 
loi  6 
114.3 


123 
24.6 

369 
49.2 

615 

73-8 

86. 1 

"•4 


138 
27  6 
41.4 
552 
69  o 
82.8 
966 
1 10.4 
124  2 


145 
39.0 
43  5 
58.0 
725 
870 
101.5 
1160 
130.5 


14  I 
28.2 
423 
564 
70-5 
84.6 
98.7 
112.8 
126.9 


137 

137 
27.4 
41. 1 

54.8 
68.5 
82.2 

95  9 
109.6 
123  3 


14-4 
288 

576 
72  o 
864 
X00.8 
115.3 
129.6 

140 

14.0 
280 
42.0 
56  o 
70.0 
84.0 
980 
112.0 
126.0 


136 

13-6 
27  2 
40.8 
54  4 
68.0 
81  6 
95-2 
108.8 


134        133       132 


13-4 
26.8 
40.2 
.  53.6 
67  o 
80.4 
93-8 
107  2 
120.6 


130 

130 
26.0 

390 
52.0 
65.0 
78.0 
91. o 
104.0 
117.0 


126 

12.6 
25  2 
37.8 
50-4 
63  o 
75-6 
88.2 
100.8 
II3-4 


12.2 
24.4 
36.6 
48.8 
61.0 
73-2 
854 
97.6 


133 
26.6 

39  9 
53-2 
665 
798 
93  I 
106  4 
119.7 


129 

12.9 
258 
387 
516 
64.5 
77-4 
90-3 
103  2 
116.1 


125 
12.5 
25  o 
37  5 
50  o 
62  5 
750 
87.5 
100  o 
112.5 


12. t 
24  2 

363 
484 
605 
72.6 
847 
96.8 


13-2 
26.4 
396 
52.8 
660 
79.2 
92  4 

105  6 


128 

128 
25  6 
38.4 
51  2 
64.0 
76.8 
89.6 
102  4 
115-2 

124 
12.4 
24.8 
37  2 
496 
62  o 
74-4 
86  8 
99.2 
111.6 


12  o 
24.0 
36  o 
48.0 
60.0 
72.0 
840 
96.0 


110.7     109.8     108  9     108.0 


P.  P. 


Qy1° 


32 


■^ 


2 

3 

4 
5 
6 

7 
8 

9 

10 

II 

12 

13 
14 
15 

16 

17 

18 

19 
20 

21 

22 
23 
24 

25 
26 

27 
28 
29 

30 

31 

32 
33 
34 
35 
36 

2>7 
38 
39 
40 

41 
42 

43 
44 
45 
46 

47 
48 

49 

50 

51 

52 
53 

54 
55 
56 

57 
58 
59 
60 


L.  Sin. 

9.01  923 


9.02  043 
9.02  163 
9.02  283 
9.02  402 
9.02  520 
9.02639 
9.02  757 
9.02  874 
9.02  992 


903  109 
9.03  226 
9-03  342 
9-03  458 

903574 
9.03  690 
9.03  805 

9.03  920 

9.04  034 
9.04  149 


9.04  262 


9.04  376 
9.04  490 
9.04  603 
9.04715 
9.04  828 

9.04  940 

9.05  052 
9.05  164 
9-05  27? 


9-05  386 

9-05  497 
9.05  607 
9.05  717 

9.05  827 

905  937 

9.06  046 

9.06155 
9.06  264 
9.06  372 


9.06481 


9.06  589 
9.06  696 
9.06  804 

9.06  911 
9.07018 

9.07  124 
9.07  231 
907  337 
9.07  442 


9-07  548 
9-07  653 
9.07  758 
9.07  863 

9.07  968 

9.08  072 
9.08  176 
9.08  280 
9.08  383 
9.08  486 


9.08  589 
L.  Cos. 


i«^1 


L.  Tan. 

9.02  162 


9.02  283 
9.02  404 
9.02  525 
9.02  645 
9.02  766 

9.02  885 

9.03  005 
9.03  124 
9.03  242 


903  361 
9.03  479 
9-03  597 
9.03  714 

9.03  832 

9.03  948 

9.04  065 
9.04  181 
9.04  297 
9.04413 


9.04  528 

9.04643^ 
9.04  758 
9.04  873 

9.04  987 

9.05  lOI 
9.05  214 
9.05  328 
9.05  441 
9-05  553 


9.05  666 

9-05  778 

9.05  890 

9.06  002 
9.06  113 
9.06  224 
906  335 

1.06  445 
9.06  556 
9.06  666 
9-o6  775 


9.06  885 

9.06  994 

9.07  103 
9.07  211 
9.07  320 
9.07  428 
9.07  536 
9.07  643 
9.07751 


9.07  858 

9.07  964 
9.08071 
9.08177 

9.08  283 
9.08  389 
9.08  495 
9.08  600 
9.08  705 
9.08  810 


9.08914 
L.  Cot. 


c.d. 

21 
21 
21 
20 
21 
19 
20 

19 
i8 

19 
18 
18 
17 
18 
16 

17 
16 
16 
16 
15 
15 
15 
15 
14 
14 
13 
H 

13 
12 

13 
12 
12 

12 
II 
II 
II 

10 
II 
10 
09 
10 
09 
09 
08 
09 
08 
08 
07 
08 
07 
06 
07 
06 
06 
06 
06 

05 
05 
05 
04 

cTd! 


L.  Cot. 

0.97  838 


0.97717 

o  97  596 
0.97  475 

0-97  355 
0.97  234 
0.97115 
0.96  995 
0.96  876 
0.96  758 


0.96  639 


0.96  521 
0.96  403 
o  96  286 
0.96  168 
0.96052 
0-95  935 
0.95  819 
0.95  703 
0-95  587 


095472 


0.95  357 
0.95  242 
0.95  127 
0.95013 
0.94  899 
0.94  786 
0.94  672 

0.94  559 
0.94  447 


0-94  334 


0.94  222 
0.94  no 
0.93  998 

0.93  887 
0.93  776 
0.93  665 

0.93  555 
0.93  444 
0.93  334 


0.93  225 


0.93115 
0.93  006 
0.92  897 
0.92  789 
0.92  680 
0.92572 
0.92  464 
0.92  357 
0.92  249 


0.92  142 


0.92  036 
0.91  929 
0.91  823 
0.91  717 
0.91  611 
0.91  505 
0.91  400 
0.91  295 
0.91  190 


0.91  086 
L.  Tan. 


L.  Cos. 

9»9"  761 
9.99  760 
9-99  759 
9.99  757 
9-99  755 
9-99  755 
9-99  753 
9-99  752 
9-99751 
9.99  749 


9-99  748 


9-99  747 
9-99  745 
9.99  744 

9.99  742 
9.99  741 
9.99  740 

9-99  738 
9-99  737 
9.99  73(> 


9-99  734 


9-99  1Z2> 
9-99  73' 
9-99  730 
9.99  728 
9.99  727 
9.99  726 
9.99  724 
9-99  723 
9-99  72' 
9.99  720 


9.99718 
9.99717 
9.99  716 
9.99714 

9-99713 
9.99711 

9.99  710 
9.99  708 
9.99  707 


■99  705 


9.99  704 
9.99  702 
9.99  701 
9.99  699 
9.99  698 
9.99  696 
9.99  695 
9-99  693 
9-99  692 


9.99  690 


9.99  689 
9.99  687 
9.99  686 
9.99  684 
9.99  683 
9.99  681 
9.99  680 
9.99  678 
9.99  677 


^99  675 
L.  Sin. 


60 

59 
58 
57 
56 
55 
54 

53 
52 
51 
50 

49 
48 
47 
46 
45 
44 

43 
42 
41 
40 

39 
^^ 
37 
36 

35 
34 

Z2> 
32 
31 
30 

29 

28 

27 
26 
25 
24 

23 
22 
21 

20 

19 
18 

17 
16 
15 
14 

13 
12 

10 

9 

8 

7 
6 
5 
4 

3 
2 
I 

O 


P.P. 


121 

120 

119 

118 

I 

12. 1 

12.0 

j;:i 

11.8 

2 

24.2 

24.0 

2^.6 

3 

36.3 

36.0 

35.7 

^S-4 

4 

4«.4 

48.0 

47-b 

47.2 

5 

bo.5 

60.0 

59-5 

59-0 

b 

72.6 

72.0 

71.4 

70.8 

7 

«4-7 

84.0 

«S.^ 

82.6 

« 

96.8 

96.0 

95.2 

94.4 

9 

108.9 

108.0 

107. 1 

106.2 

117 

116 

115 

114 

I 

11.7 

11.6 

"•5 

II. 4 

2 

23.4 

23.2 

23.0 

22.8 

3 
4 

J?:s 

34.8 
46.4 

34.5 
46.0 

34-2 
4,S.6 

5 
6 

7 

58.5 
70.2 
81.9 

58.0 
69.6 
81.2 

69.0 
80.5 

79.8 

8 

93-6 

92.8 

92.0 

91.2 

9 

105.3 

104.4 

103-5 

102.6 

"•3 
22.6 
33-9 
45-2 
56.5 
67.8 
79.1 
90.4 
101.7 


109 

10.9 
21.8 
32.7 
43-6 
54-5 
65-4 
76.3 
87.2 


22.4 
33-6 
44.8 
56.0 
67.2 
78.4 
89.6 
100.8 


108 

10.8 
21.6 
32.4 
43.2 
54-0 
648 
75-6 
86.4 
97.2 


22.2 

33-3 
44.4 
55-5 
66.6 
77-7 


107 

10.7 
21.4 
32.1 
42.8 
53-5 
64.2 

74-9 
85.6 
96.3 


22.0 
33-0 
44.0 
55-0 
66.0 
77.0 


106 
10.6 
21.2 
31.8 
42.4 

53-0 
63.6 
742 
84.8 
95-4 


105 

1  I  10.5 

2  I  21.0 

3  j  31-5 

4  i  42.0 

5  j  52.5 

63.0 
73-5 
84.0 
94-5 


104 

10.4 
20.8 
31.2 
41.6 
52.0 
62.4 
72.8 
83.2 
93-6 


P.  P. 


103 

10.3 
20.6 

309 
41.2 
51-5 
61.8 
72.1 
82.4 
92.7 


•    83° 


33 


/ 

L.  Sin. 

d. 

L.  Tan. 

c.d.  L.  Cot. 

L.  Cos. 

P.  P. 



0 

I 

9.08  589 

103 

9^08914 
9.09019 

105 

0.91  086 

9.99  b75 

60 

59 

9.08  692 

0.90981 

9-99  674 

2 

9.08  795 

I  "3 

9.09123 

104 

0.90  877 

9-99  ^>72 

58 

105 

104 

103 

3 

9.08  897 

9.09  227 

104 

0.90  773 

9.99  670 

57 

I 

10.5 

10.4 

10.3 

4 

9.08  999 

9-09  330 

103 

0.90  670 

9-99  (^('9 

5^' 

2 

21.0 

20.8 

20.6 

5 

9.09  10 1 

9-09  434 

103 

0.90  566 

9.99  667 

55 

3 

315 

31.2 

30-9 

6 

9.09  202 

909  537 

0.90  463 

9.99  666 

54 

4 

42.0 

41.6 

41.2 

7 

9.09  304 

9.09  640 

103 

0.90  360 

9.99  664 

S3 

5 

52-5 

52.0 

5^-5 

8 

9.09  405 

9.09  742 

0.90  258 

9.99  663 

52 

6 

63.0 

62.4 

61.8 

9 
10 

9.09  500 
9.09  606 
9^9707 

100 

lOI 

9.09  845 
9.09  947 
9.10049 

103 
102 
102 

0.90  155 

9.99  661 

5^ 
50 

49 

7 
8 

9 

73-5 
84.0 

94-5 

72.8 
83.2 
93-6 

72.1 
82.4 
92.7 

0.90053 

9-99  659 

0.89951 

9.99  658 

12 

9.09  807 

9.10  150 

0.89  850 

9-99  656 

48 

13 

9.09  907 

9.10  252 

0.89  748 

9-99  655 

47 

H 

9.10006 

99 

9-10353 

lOl 

101 

0.89  647 

9-99  653 

46 

102 

101 

99 

^S 

9.10  106 

9.10454 

0.89  546 

9.99651 

45 

I 

10.2 

10.1 

9-9 

lb 

9.10  205 

99 

9-10555 

0.89  445 

9-99  650 

44 

2 

20.4 

20.2 

19.8 

17 

9.10304 

99 

98 

9.10656 

0.89  344 

9.99  648 

43 

3 

30.6 

30.3 

29.7 

18 

9.10  402 

9.10756 

0.89  244 

9-99  647 

42 

4 

40.8 

40.4 

39-6 

19 
20 

21 

22 

9.10  501 

99 
98 

98 
98 
98 

9.10856 

100 

100 

99 

0.89  144 

9-99  645 

41 
40 

39 
38 

5 
6 

51.0 
61.2 
71.4 
81.6 

50-5 
60.6 
70.7 
80.8 
90.9 

49-5 

79-2 
89.1 

9.10599 

9.10956 

0.89  044 

9.99  643 

9.10697 
9.10795 

9.11  056 

9"  155 

0.88  944 
0.88  845 

9.99  642 
9.99  640 

23 

9.10893 

9.1 1  254 

99 

0.88  74b 

9.99  638 

37 

9    y..^ 

24 

9.10990 

97 

9-II353 

99 

0.88  647 

9.99  637 

36 

2S 

9. 1 1  087 

97 

9.11452 

99 

0.88  548 

999635 

35 

26 

9.1 1  184 

97 

9-11551 

99 
98 
98 
98 
98 

97 
98 
97 
97 
96 

0.88  449 

9-99  (>33 

34 

98 

97 

96 

27 

9.11  281 

9; 
96 

9. 1 1  649 

0.88351 

9.99  632 

33 

I 

9-8 

9-7 

9.6 

28 

9-"  377 

9.11  747 

0.88  253 

9-99  630 

32 

2 

19.6 

19.4 

19.2 

29 

30 

31 
32 

33 
34 

9. 1 1  474 

9y 
96 

96 

95 
96 

95 

9.11  845 

0.88155 

9.99  629 

31 
30 

29 
28 
27 
26 

3 

4 
5 
6 

7 
8 
9 

29.4 
39.2 
49.0 
58.8 
68.6 

29.1 
38.8 
48.5 
58.2 
67.9 
77.6 
87-3 

28.8 

38.4 
48.0 

57-6 
67.2 
76.8 
86.4 

9.11  570 

9. 1 1  943 

0.88057 

9.99  627 

9.11  666 
9.1 1  761 
9.11857 
9.11952 

9. 1 2  040 
9.12  138 
9.12235 
9.12332 

0.87  960 
0.87  862 
0.87  765 
0.87  668 

9.99  625 
9.99  624 
9.99  622 
9.99  620 

3S 

9.12047 

95 

9.12428 

0.87572 

9.99  6x8 

25. 

36 

9.12  142 

95 

9.12525 

97 
96 
96 
96 
96 

95 

0.87  475 

9.99617 

24 

37 

9.12236 

94 

9.12  621 

0.87  379 

9.99615 

23 

95 

94 

93 

3« 

9-12331 

95 

9.12717 

0.87  283 

9.99613 

22 

39 

9.12425 

94 

9.12813 

0.87  187 

9.99  612 

21 

I 

9-5 

9-4 

9-3 

40 

41 

9.12  519 

94 
93 

9.12909 

0.87091 

9.99  610 

20 

19 

2 
3 
4 
5 
6 

19.0 

28.5 
38.0 

47-5 
57-0 

18.8 
28.2 
37.6 
47.0 

56.4 

18.6 
27.9 
37-2 
46.5 
55-8 

9.12612 

9.13004 

0.86  996 

9.99  608 

42 

43 

9.12  706 
9.12799 

94 
93. 
93 

9.13099 
9-13  194 

95 
95 
95 

0.86  901 
0.86  806 

9.99  607 
9.99  605 

18 
17 

44 

9.12  892 

9.13289 

0.86  711 

9-99  603 

16 

7 

66.S 

65.8 

65.1 

4S 

9.12985 

93 

9- 1 3  384 

95 

0.86616 

9.99  601 

IS 

8 

76.0 

75-2 

74-4 

46 

9.13078 

93 

9-13478 

94 
95 

0.86  522 

9.99  600 

14 

9 

85.5 

84.6 

83-7 

47 

913  171 

93 

9-13573 

0.86427 

9-99  598 

13 

48 

9.13263 

92 

9.13667 

94 

0.86  333 

9-99  596 

12 

49 
50 

913355 

92 
92 
92 

9.13761 

94 
93 
94 
93 

0.86 146 

9-99  595 

11 
10 

QQ 

91 

9.1 
18  2 

90 

9-13447 

9-13854 

9-99  593 

I 
2 

51 

9-13539 

9.13948 

0.86052 

9-99591 

9 

9-2 

18.4 
27.6 
36.8 

9.0 
180 

S2 

9.13630 

91 

9. 14  041 

0.85  959 

9-99  589 

8 

3 
4 

27.3 
36.4 

27.0 
36.0 

53 

9.13722 

92 
91 

9.14134 

93 
93 
93 

0.85  866 

9.99  588 

7 

54 

9-13813 

9.14227 

0.85  773 

9-99  586 

6 

S 

46.0 

45-5 

45 -o 

55 

9.13904 

91 

9.14320 

0.85  680 

9-99  584 

s 

6 

55-2 

54-6 

54.0 

5^ 

9-13994 

90 

9.14412 

92 

0.85  588 

9.99  582 

4 

7 

64.4 

63-7 

63.0 

S7 

9.14085 

91 

9.14504 

92 

0.85  496 

9.99581 

3 

8 

73.6 

72.8 

72.0 

5« 

914  175 

90 

9-14597 

93 
91 
92 

0.85  403 

9-99  579 

2 

9 

82.8 

81.9 

81.0 

59 
60 

9.14266 

91 
90 

9.14688 

0.85  312 

9-99  577 

0 

9.14356 

9.14780 

0.85  220 

9-99  575 

_ 

L.  Cos. 

d. 

L.  Cot. 

c.d.|  L.  Tan. 

L.  Sin. 

/ 

P.P.          1 

82' 


34 


'  1  L.  Sin.  1  d. 

L.  Tan.  |c.d.  |  L.  Cot.  |  L.  Cos.  | 

P.  P. 

O 

I 

9.14356 

89 
90 
89 
90 

.89 
8X 

9.14  780 

92 

0.85  220 

9-99  575 

60 

59 

9-14  445 

9.14872 

0.85  128 

9.99  574 

2 

9-14  535 

9.14963 

91 
91 
91 

0.85  037 

9.99  572 

S8 

92    91    90 

3 

9.14  624 

9-15054 

0.84  946 

9.99  570 

57 

I 

9.2    9.1    9.0 

4 

9.14714 

9.15  145 

0.84  855 

9-99  568 

56 

2 

18.4   18.2   18.0 

5 

9.14803 

9.15  236 

91 
91 

0.84  764 

9-99  566 

55 

3 

27.6   27.3   27.0 

6 

9.14  891 

89 
89 
88 
88 
88 
88 

9.15327 

0.84  673 

9.99  565 

54 

4 

36.8   36.4   36.0 

7 

9.14980 

9.15  417 

90 

0.84  583 

9.99  563 

53 

5 

46.0   45.5   45.0 

b 

9.15069 

9.15  508 

91 

0.84  492 

9-99  561 

52 

6 

55.2   54.6   540 

9 
10 

II 

915  157 

9.15  598 

90 
90 
89 
90 
89 

0.84  402 

9.99  559 

51 
50 

49 

7 
8 

9 

64.4   63.7   63.0 
73.6   72.8   72.0 
82.8   81.9   81.0 

9-15245 

9.15  688 

0.84  3 1 2 

9.99  557 

9-15333 

9.15777 

0.84  223 

9-99  556 

12 

9.15421 

87 
88 

87 
87 
87 

87 
86 
86 

87 
86 

9.15867 

0.84  133 

9.99  554 

48 

13 

9.15508 

9.15956 

0.84  044 

9.99552 

47 

H 

9-15596 

9.16046 

90 
89 
89 
88 

89 
88 
88 
88 
88 

0.83  954 

9-99  550 

46 

89     88 

»5 

9.15683 

9.16135 

0.83  865 

9.99  548 

4S 

I 

8.9    8.8 

10 

9.15770 

9.16224 

0.83  776 

9.99  546 

44 

2 

17.8    17.6 

^7 

9-15857 

9.16312 

0.83  688 

9-99  54? 

43 

3 

26.7   26.4 

i8 

9-15944 

9.16401 

0.83  599 

9-99  543 

42 

4 

35-6   35.2 

19 
20 

21 

9.16  OJO 

9.16489 

0.83  51 1 

9.99  541 

41 
40 

39 

7 
8 

44.5   44.0 
53-4   52.8 
62.3   61.6 
71.2   70.4 
80.1    79.2 

9.16  116 

9.16577 

0.83  423 

9.99  539 

9.16  203 

9.16  665 

0.83  335 

9.99  537 

22 
23 

9.16  289 
9.16374 

85 
86 

85 
86 

85 
85 
85 
84 
85 
84 
84 
84 
84 
83 
84 
S3 
83 
83 
83 

83 
82 

82 

9-16753 
9.16  841 

88 

87 
88 

87 

87 

87 
86 

87 
86 
86 

0.83  247 
0.83  159 

9.99  535 
9.99  533 

38 
37 

9 

24 

9.16460 

9.16928 

0.83  072 

9.99  532 

36 

25 

9.16545 

9.17016 

0.82  984 

9-99  530 

3S 

25 

9-i6  631 

9.17  103 

0.82  897 

9.99  528 

34 

87    86    85 

27 

9.16  716 

9.17  190 

0.82810 

9-99  526 

3^ 

I 

8.7    8.6    8.5 

28 

9.16  801 

9.17277 

0.82  723 

9.99  524 

32 

2 

17.4   17.2   17.0 

29 

30 

31 

9.16886 

9.17363 

0.82  637 

9.99  522 

31 
30 

29 

3 
4 

5 
6 

26.1  25.8   25.5 

34.8  34.4   340 

43.5  43.0   42.5 

52.2  51.6   51.0 

60.9  60.2   59.5 

69.6  68.8   68.0 

9.16970 

9-17450 

0.82  550 

9.99  520 

9-17055 

9- '7  536 

0.82  464 

9.99518 

32 

9.17  139 

9.17622 

86 

0.82  378 

9-99517 

28 

7 

ss 

9.17223 

9.17708 

86 

0.82  292 

9.99515 

27 

8 

34 

9.17307 

9.17794 

86 

0.82  206 

9.99513 

26 

9 

78.3   77-4   76.5 

35 

9-1739' 

9.17880 

85 
86 

85 
85 
85 
85 
84 
85 
84 
84 
84 

84 
83 
84 
83 
83 

^3 
83 

82 
82 
82 
82 
82 

0.82  120 

9.99511 

25 

3b 

9.17474 

9.17965 

0.82  035 

9-99  509 

24 

37 

9-17558 

9.18051 

0.81  949 

9-99  507 

23 

84     83 

3^ 

9.17641 

9.18136 

0.81  864 

9-99  505 

22 

39 
40 

41 
42 
43 

9.17724 

9.18  221 

0.81  779 

9.99  503 

21 
20 

18 
17 

I   " 

2 

3 

4 

5 

6 

0.4    0.3 
16.8    16.6 
25.2    24.9 
33.6   33.2 
42.0   41.5 
50.4   49.8 

9.17807 

9.18  306 

0.81  694 

9-99  501 

9.17890 

9.17973 
9.18055 

9.18391 

9.18475 
9.18560 

0.81  609 
0.81  525 
0.81  440 

9.99  499 
9.99  497 
9-99  495 

44 
45 

9.18137 
9.18  220 

82 
81 
82 
82 
81 
81 
81 
81 
81 

9.18644 
9.18728 

0.81  356 
0.81  272 

9.99  494 
9.99  492 

16 
IS 

I 

58.8   58.1 
67.2   66.4 

4t) 

9.18302 

9.18812 

0.81  188 

9.99490 

14 

9 

75-6   74-7 

47 

9.18383 

9.18896 

0.81  104 

9.99  488 

n 

48 

9.18465 

9.18979 

0.81  021 

9  99  486 

12 

49 
50 

5' 

9-18547 

9.19063 

0.80  937 

9-99  484 

II 
10 

9 

fio.    fii    fin 

9.18628 

9.19  146 

0.80  854 

9.99482 

I 
2 

8.2    8.1    8.0 
16.4   16.2   16.0 
24.6   24.3   24.0 
32.8   32.4   32.0 

9.18709 

9.19229 

0.80771 

9.99  480 

52 
53 

9.18790 
9.18871 

9.19312 
9.19395 

0.80  688 
0.80  605 

9.99  478 
9.99  476 

8 
7 

3 

4 

54 

918952 

Rj 

9.19478 

0.80  522 

9.99  474 

6 

41.0   40.5   40.0 

55 

9-19033 

80 
80 
80 
80 
80 

919561 

0.80  439 

9.99472 

5 

6 

49.2   48.6   48.0 

5^ 

9.F9113 

9.19  643 

0.80357 

9.99  470 

4 

7 

57-4   56.7   56.0 

57 

9.19  193 

9.19725 

0.80  275 

9.99  468 

3 

8 

65.6   64.8   64.0  • 

5« 

9  19273 

9.19807 

0.80  193 

9.99  466 

2 

9 

73.8   72-9   720 

59 

919353 

9.19889 

0.80  1 1 1 

9.99  464 

I 

60 

919433 

9.19971 

0.80  029 

9.99  462 

0 

L.  Cos. 

d. 

L.  Cot. 

c.d. 

L.  Tan. 

L.  Sin. 

'          P.  P.         1 

sr 


35 


'  j  L.  Sin.  1  d.  1  L.  Tan. 

c.d.  1  L.  Cot.  1  L.  Cos.  1 

P.P.         1 

0 

9- 19  433 

80 

9.19971 

82 
81 
82 
81 
■  81 
81 
81 
81 
80 
81 
80 
80 
80 
80 
80 

0.80  029 

9.99  462 

60 

I 

9'95«3 

9.20053 

0.79947 

9.99  4O0 

59 

2 

3 

9.19592 
9.19672 

79 
80 

9.20  134 
9.20  216 

0.79  866 
0.79  784 

9.99  458 
9.99  456 

58 
57 

J 

82 

8.2 

81 

8.1 

80 

8.0 

4 

9-'9  75i 

79 

9.20  297 

0.79703 

9-99  454 

56 

2 

16.4 

16.2 

16.0 

5 

9.19830 

79 

9.20378 

0.79  622 

999452 

55 

3 

24.6 

24-3 

24.0 

6 

9.19909 

79 

9.20459 

0.79  541 

9.99  450 

54 

4 

32-8 

32.4 

32.0 

7 

9.19988 

;9 

9.20  540 

0.79  460 

9.99  448 

53 

5 

41.0 

40.5 

40.0 

8 

9.20067 

V9 
78 
78 
79 
78 
78 

9.20621 

0.79379 

9.99  446 

52 

6 

49-2 

48.6 

48.0 

9 
10 

II 

9.20  145 

9.20  701 

0.79  299 

9.99  444 

51 
50 

49 

7 
8 

9 

57-4 
65.6 
73-8 

64.8 
72.9 

56.0 
64.0 
72.0 

9.20223 
9.20  302 

9.20  782 

0.79  218 

9.99  442 

9.20  802 

0.79  138 

9.99  440 

12 

9.20380 

9.20942 

0.79058 

9.99  438 

48 

^3 

9.20458 

9.21  022 

0.78978 

9-99  436 

47 

79 

78 

77 

14 

920  535 

78 

9.21  102 

0.78  898 

9-99  434 

46 

I 

7.9 

7.8 

7-7 

«S 

9.20613 

9.21  182 

0.78818 

9-99  432 

45 

2 

15.8 

15-b 

15-4 

i6 

9.20691 

9.21  261 

79 
80 

0.78  739 

9.99  429 

44 

3 

237 

23.4 

23-1 

I? 

9.20768 

n 

9.21  341 

0.78659 

9.99  427 

43 

4 

3i.t> 

31.2 

30.8 

i8 

9.20  845 

'I'i 

9.21  420 

;9 

0.78  580 

9-99  425 

42 

5 

39.5 

39-0 

38.5 

19 

9.20922 

ri 

9.21  499 

79 

0.78501 

9-99  423 

41 

6 

47-4 

46.8 
54.6 
62.4 
70.2 

46.2 

20 

21 

9.20999 

77 
77 

9.21  578 

79 
79 

0.78422 

9.99421 

40 

39 

i 

9 

55-3 
63.2 
71. 1 

53-9 
61.6 

69.3 

9.21  076 

9.21  657 

0.78  343 

9.99419 

22 

9-21  153 

77 
76 

9-21  736 

7^ 
78 

0.78:^64 

9.99417 

38 

23 

9.21  229 

9.21  814 

0.78  186 

9-99415 

37 

24 

9.21  306 

76 
76 
76 
76 

9-21  893 

79 

7? 
78 

78 

78 

78 

78 

77 

78 

0.78  107 

9-99413 

36 

76 

75 

74 

2,S 

9.21  382 

9.21 971 

0.78029 

9.99411 

35 

I 

7.6 

7-5 

7.4 

26 

9.21  458 

9.22  049 

0.77951 

9-99  409 

34 

2 

15-2 

15.0 

14.8 

27 

9-21  534 

9.22127 

0.77873 

9.99  407 

33 

3 

22.8 

22.5 

22.2 

28 

9.21  610 

9.22  205 

0.77  795 

9.99  404 

32 

4 

30.4 

30.0 

29.6 

29 

30 

31 

9.21  685 

75 
76 

75 
76 

9.22  283 

0.77717 

9.99  402 

31 
30 

29 

I 

38.0 
45-6 
53-2 
60.8 

37.5 
45-0 

60.0 

37-0 
66!6 

9.21  761 

9.22361 

0.77  639 

9.99  400 

9.21  836 

9.22438 

0.77  562 

9.99  398 

32 

9.21  912 

9.22  516 

0.77484 

9.99  396 

28 

9 

68.^ 

67.5 

^2, 

9.21  987 

75 

9.22  593 

77 

0.77  407 

9-99  394 

27 

34 

9.22062 

Vb 

9.22  670 

77 

0.77  330 

9.99  392 

26 

73 

72 

71 

35 

9.22  137 

75 

9.22  747 

77 

0.77  253 

9-99  390 

25 

3b 

9.22  211 

74 

9.22  824 

77 

0.77  176 

9.99388 

24 

I 

7.3 

7.2 

7-1 

37 

9.22  286 

75 

9.22  901 

77 
76 

0.77  099 

9-99  385 

23 

2 

14.6 

14.4 

14.2 

38 

9.22  36  F 

75 

9.22977 

0.77  023 

9-99  383 

22 

3 

21.9 

21.6 

^«-^ 

39 
40 

41 

9.22  435 

74 
74 
74 

923054 

76 
76 

0.76  946 

9-99  381 

21 
20 

19 

4 

7 
8 

29.2 

36.5 
43-8 
5I-I 
58.4 

28.8 
36.0 
43.2 
50.4 

57.6 
64.8 

28.4 

42.6 

497 
56.8 

639 

9.22  509 

9.23  130 

0.76  870 

9-99  379 

9.22  583 

9.23  200 

0.76  794 

9-99  377 

42 

9.22657 

lA 

9.23  283 

76 

0.76717 

9.99  375 

18 

43 

9.22731 

74 

9-23  359 

0.76  641 

9-99  372 

17 

9 

65.7 

44 

9.22  805 

74 

923435 

0.76  565 

9-99  370 

16 

45 
46 

9.22878 
9.22952 

73 

9.23  510 
9.23  586 

75 

0.76  490 
0.76414 

9-99  368 
9  99  366 

^5 
14 

74 

t 

47 

9.23025 

73 

9.23  661 

75 
76 

0.76  339 

9-99  364 

13 

3 

3 

3 

48 

9.23098 

73 

923737 

0.76  263 

9.99  362 

12 

79 

78 

77 

49 
50 

5« 

9.23  171 
9.23  244 
9.23317 

IS 

9.23  812 

lb 

0.76  188 

9-99  359 

10 

9 

0 
I 
2 
3 

13.2 
65!8 

13.0 
39.0 
65.0 

12.8 

38.5 
64.2 

9.23  887 

V5 
75 

0.76  113 

9-99  357 

9.23962 

0.76038 

9.99  355 

52 

9.23  390 

73 

9.24037 

75 

0.75  963 

9-99  353 

8 

53 

9.23462 

72 

9.24  112 

IS 

0.75  888 

999351 

7 

3 

3 

3 

54 

9- 23  53? 

73 

9.24  186 

74 

0.75  814 

9.99  348 

6 

1R 

75 

74 

55 

9.23  607 

72 

9.24261 

75 

0.75  739 

9-99  346 

5 

0 
I 

5^ 

9^.23  679 

72 

9.24  335 

H 

0.75  665 

9-99  344 

4 

12.7 

12.5 

12.3 

57 

923752 

73 

9.24410 

75 

0.75  590 

9.99  342 

3 

2 

38.0 

37-5 

37-0 

5« 

9.23823 

yi 

9.24  484 

74 

075516 

9.99  340 

2 

3 

^6-3 

62.5 

61.7 

59 

9.23895 

72 
72 

9-24558 
9.24632 

;4 
74 

0.75  442 

9-99  337 

I 
0 

60 

9.23967 

0.75  368 

9-99  335 

1  L.Cos. 

1  d. 

L.  Cot. 

|c.d.|  L.  Tan. 

L.  Sin.  1  ' 

1 

P.P. 

80^ 


36 


j  / 

L.  Sin. 

d. 

L.  Tan. 

c.  d. 

L.  Cot. 

L.  Cos. 

d. 

1      P.P.      Il 

0 

I 

9-23967 

72 

71 
71 

72 

71 
71 

9.24  632 

74 
73 
74 
73 
74 
73 

0.75  368 

9-99  335 

2 

60 

59 

9.24  039 

9.24  70b 

0.75  294 

9.99  333 

2 

9.24  no 

9.24  779 

0.75  221 

9-99  331 

58 

74  73   72 

3 

9.24  181 

9-24  853 

0-75  147 

9-99  328 

3 

57 

I 

7.4  7-3  7-2 

4 

9-24  253 

9.2492b 

0.75  074 

9-99  326 

56 

2 

14.8  14.6  14.4 

5 

9.24  324 

9.25  000 

0.75  000 

9.99  324 

2 

55 

3 

22.2  21.9  2i.b 

6 

9-24  395 

9-25  073 

0.74927 

9-99  322 

54 

4 

29.b  29.2  28.8 

7 

9.24  46b 

7^ 
70 

71 
70 

71 
70 
70 
70 
70 
70 

9.25  14b 

73 

0.74  854 

9-99  319 

3 

53 

c 

37.0  3b.5  3b.o 

8 

9.24  536 

9.25219 

73 
73 

73 

72 

73 
72 

73 
72 
72 

0.74  781 

9-99317 

2 
2 
2 

3 

52 

6 

44.4  43-8  43-2 

9 
lO 

II 

9.24  boy 

9-25  292 

0.74  708 

9-99315 

51 
50 

49 

9 

51.8  51. 1  50.4 
59-2  584  57-6 
bb.6  65.7  b4.8 

9.24  b77 

9.25  3^5 

0.74  635 

9-99  3'3 

9.24  748 

9-25  437 

0.74  563 

9.99310 

12 

9.24818 

9-25510 

0.74  490 

9-99  308 

48 

13 

9.24  888 

9-25  582 

0.74418 

9.99  306 

2 

3 

2 

47 

71   70   69  1 

14 

9.24958 

9-25  655 

074  345 

9-99  304 

46 

I 

7.1  7.0  b.9 

15 

9.25  028 

9.25  727 

0.74  273 

9-99  301 

45 

2 

14.2  14.0  13.8 

lb 

9.25  098 

9-25  799 

0.74  201 

9-99  299 

44 

3 

21.3  21.0  20.7 

17 

9.25  ib8 

70 

69 
70 

69 
69 
69 
69 
60 

9.25871 

72 
72 
71 
72 

71 

72 

71 
71 
71 

0.74  129 

9-99  297 

2 

3 

43 

4 

28.4  28.0  27.b 

18 

9-25  237 

9-25  943 

0.74057 

9.99  294 

42 

5 
6 

7 
8 

9  1 

35-5  350  34-5 

42.6  42.0  41.4 

49.7  49-0  48.3 
5b.8  56.0  55.2 
bi.Q  b^.o  b2.i 

19 

20 

21 

925  307 

9.26015 

0.73  985 

9.99  292 

2 
2 
3 

41 
40 

39 

9-25  376 

9.26  086 

0.73914 

9.99  290 

9-25  445 

9.26  158 

0.73  842 

9.99  288 

22 

9-25  5H 

9.26  229 

0.73  771 

9.99  285 

38 

23 

925  583 

9.26  301 

0.73  b99 

9.99  283 

2 

37 

68  67   66 

24 

9.25  b52 

69 
69 
b8 

69 
b8 

9-26372 

0.73  628 

9.99  281 

3 

36 

25 

9.25  721 

9-26443 

0-73  557 

9.99  278 

35 

I 

b.8  b.7  6.b 

2b 

9.25  790 

9-26514 

0.73  48b 

9-99  276 

34 

2 

13.6   13.4  13.2 

27 
28 

9  25  858 
9.25927 

9-26  585 
9-26655 

70 
71 
71 
70 
70 

71 
70 
70 
70 
70- 
69 

r 
69 

70 

69 
69 

69 
69 
69 

69 
69 

68 

0-73415 
0-73  345 

9.99  274 
9.99  271 

3 

2 
2 

3 
2 

33 
32 

3 
4 

20.4  20.1  19.8 
27.2  2b.8  2b.4 

29 

30 

31 

9-25  995 

68 
68 
68 

9.26  726 

0.73  274 

9.99  269 

31 
30 

29 

I 

34-0  33-5  33-0 
40.8  40.2  39.b 
47. b  46.9  4b.2 
54.4  53.b  52.8 

9.2b  ob3 

9.26  797 

0.73  203 

9.99  267 

9.2b  131 

9.2b  8b7 

0.73  ^33 

9.99  2b4 

32 

9.2b  199 

68 

9-26937 

0.73  ob3 

9.99  2b2 

28 

9 

bi.2  bo.3  59.4 

33 

9.26  2b7 

68 

9.27  008 

0.72992 

9.99  2bO 

3 

27 

1 

34 

9-26335 

68 

9.27078 

0.72922 

9-99  257 

2b 

65   3 

35 

9.2b  403 

67 
68 

67 
67 
67 
67 
67 
67 
67 
66 

67 
6b 

67 
66 

9.27  148 

0.72852 

9-99  255 

3 
2 
2 

25 

,    z:  -   „  _ 

36 

37 

9.2b  470 
926538 

9.27  218 
9.27288 

0.72  782 
0.72712 

9-99  252 
9-99  250 

24 
23 

i  "-3  'J-J 
>  1  13.0  0.6 
5  i  19-5  0.9 
[    26.0  1.2 

3H 

9.2b  bo5 

9-27357 

0.72  643 

9-99  248 

3 
2 

2 
3 

22 

39 
40 

41 

9,2b  b72 

9.27427 

0.72  573 

9-99  245 

21 

20 

19 

)  32.5  1-5 
)  39.0  1.8 
'    45-5  2.1 

9.2b  739 

9-27  496 

0.72  504 

9-99  243 

9.2b  80b 

9.27  5bb 

0.72434 

9.99  241 

42 

9-26873 

9.27  b35 

0.72  3b5 

9-99  238 

18 

I    52.0  2.4 

43 

9.2b  940 

9-27  704 

0.72  29b 

9-99  236 

3 
2 

17 

)  58-5  2-7 

44 
45 

9.27007 
9.27073 

9-27773 
9.27  842 

0.72  227 
0.72  158 

9-99  233 
9.99231 

lb 
15 

1 

4b 

47 

9.27  140 
9.27  20b 

9.27  911 
9.27  980 

0.72  089 
0.72  020 

9.99  229 
9.99  22b 

3 

2 

14 
13 

3   3   3 
74   73   72 

48 

9.27.273 

9.28049 

0.71  951 

9-99  224 

3 

2 

2 

12 

0  j 

49 
50 

9-27  339 

66 
66 

9.28  117 

69 

68 

0.71  883 

9.99221 

II 
10 

2 

12.3  12.2  12.0 
37.0  36.5  36.0 
61.7  60.8  60.0 

9.27  405 

9.28  i8b 

0.71  814 

9.99219 

51 

9.27471 

66 

9.28  254 

69 
68 

0.71  746 

9.99217 

3 

2 

9 

3 

52 

9-27  537 

65 
66 

9.28323 

0.71  b77 

9.99  214 

•  8 

53 

9.27  b02 

9.28  391 

68 

0.71  bo9 

9.99212 

3 

7 

3   3   3   3^ 

54 

9.27  6b8 

66 

9.28459 

68 

0.71  541 

9-99  209 

2 

b 

71  70  69  68 

55 

9-27  734 

65 
65 
66 

9.28527 

68 

0.71  473 

9.99  207 

3 
2 
2 

5 

0 

11.811.7  11.511.31 

35-5  35-034-5  34-Oi 
59.258.357.5  5b.7 

5^ 
57 

9.27  799 
9.27  864 

9-28  595 

9-2§.6b2 

67 
68 

0.71  405 
0-71  338 

9.99  204 
9.99  202 

4 
3 

I 
2 

5^ 

9.27  930 

65 
65 

9.28  730 

68 

0.71  270 

9.99  200 

3 

2 

2 

3 

59 
60 

9-27  995 
9.28  obo 

9.28  798 

67 

0,71  202 

9  99  197 

I 
0 

9.28  865 

0.71  135 

9-99  195 

L.  Cos. 

d. 

L.  Cot. 

c.  d.  L.Tan.  | 

L.  Sin. 

d. 

f 

P.  P. 

79' 


37 


/ 

L.  Sin. 

i  d. 

L.Tan. 

c.  d. 

L.  Cot. 

L.  Cos. 

d. 

P.  P. 

0 

I 

9.2S  060 
9.28125 

65 

65 
64 

65 
65 
64 
64 

64 
64 
64 
64 
63 
64 
64 
63 
63 
64 
63 
63 
63 
63 
63 
6? 

9.28  865 

68 

67 
67 
67 

67 
67 
67 

67 
66 

67 
66 

67 
66 

66 

66 

0.71  135 

9.99  195 

3 

60 

59 

928  933 

0.71  067 

9.99  192 

2 

9.28  190 

9.29000 

0.71  000 

9.99  190 

3 
2 

58 

68  67  66 

3 

9.28  254 

9.29067 

0.70933 

9-99  187 

57 

I 

6.8  6.7  6.6 

4 

9.28319 

9.29  134 

0.70  866 

9.99  185 

3 

56 

2 

13-6  13.4  13.2 

,s 

9.28  384 

9.29  201 

0.70  799 

9.99  182 

55 

3 

20.4  20.1  19.8 

6 

9.28  448 

9.29  268 

0.70  732 

9.99  180 

3 
2 

3 

2 

3 
2 

54 

4 

27.2  26.8  26.4 

7 

9.28512 

929  33^ 

0.70  665 

9.99177 

53 

5 

340  33-5  330 

8 

9-28577 

9.29  402 

0.70  598 

9.99175 

52 

6 

40.8  40.2  39.6 

9 
iO 

II 

9.28  b4i 
9-28  705 
9.28  769 

9.29  468 

0.70532 

9-99  172 

51 
50 

49 

7 
8 

9 

47.6  46,9  46.2 
54.4  53.6  52.8 
61.2  60.3  59.4 

9.29  535 

0.70  465 

9.99170 

9.29601 

0.70  399 

9-99  167 

12 

9.28  S33 

9.29  668 

0.70332 

9-99  1 61 

3 

48 

65  64  63 

13 

9.28896 

9.29  734 

0.70  266 

9.99  162 

47 

H 

9.28  960 

9.29  800 

0.70  200 

9.99  160 

3 

46 

I 

6.5  6.4  6.3 

^5 

9.29  024 

9.29  866 

66 

0.70  134 

9-99157 

45 

2 

13.0  12.8  12.6 

i6 

9.29087 

9.29  932 

66 
66 
66 

65 
66 

t^ 
65 

66 

0.70  068 

9-99  155 

44 

3 

19.5  19.2  18.9 

17 

9.29  150 

9.29  998 

0.70002 

9.99152 

3 

43 

4 

26.0  25.6  25.2 

18 

9.29214 

9.30  064 

0.69  936 

9-99  150 

42 

7 
8 

9 

32.5  32.0  31.5 
39.0  38.4  37.8 
45.5  44-8  44.1 
52.0  51.2  50.4 
t;8.t;  i;7.6  1:6.7 

19 
20 

21 

9.29  277 

9.30  130 

0.69  870 

9.99  147 

3 

2 

3 

41 
40 

39 

9.29  340 

930  195 

0.69  805 

9-99  145 

9.29  403 

9.30  261 

0.69  739 

9-99  142 

22 

9.29  466 

9.30  326 

0.69  674 

9-99  140 

3 

2 

38 

23 

9.29529 

9-30391 

0.69  609 

9-99  137 

37 

62  61   60 

24 

9.29591 

63 
62 

9-30457 

65 
65 
65 

64 

65 
64 
65 
64 
64 
65 
64 
64 
64 
64 

63 
64 
63 
64 

63 
64 

63 
63 
63 
63 
63 
63 
63 
62 

0.69  543 

9-99  135 

3 

36 

25 

9.29  654 

9.30522 

0.69  478 

9-99  132 

35 

I 

b.2  b.i  b.o 

2b 

9.29716 

63 
62 
62 

63 
6'> 

9.30  587 

0.69413 

9-99  130 

3 

3 

2 

3 

34 

2 

12.4  12.2  12.0 
18.6  18.3  18.0 
24.8  24.4  24.0 
31.0  30.5  30.0 
37.2  36.6  36.0 
43-4  42.7  420 
49.6  48.8  48.0 

27 
28 
-29 

30 

9  29  779 
9.29  841 
9.29  903 

9.30652 
9.30717 
9.30  782 

0.69  348 
0.69  283 
0.69  218 

9.99  127 
9-99  124 
9.99  122 

33 
32 
31 
30 

3 
4 
5 
6 

7 

9.29  966 

9.30  846 

0.69  154 

9-99  "9 

31 

9.30028 

6-' 

9.30  91 1 

0.69  089 

9.99  117 

3 
2 

29 

8 

32 

9.30  090 

6t 

9-30  975 

0.69  025 

9-99114 

28 

9 

55-8  54.9  54.0 

33 
34 

9.30151 
9.30213 

62 
62 

9.31  040 
9.31  104 

0.68  960 
0.68  896 

9.99  112 
9-99  109 

3 

3 

27 
26 

59   3 

3.S 

9-30  273 

61 

9.31  168 

0.68  832 

9.99  106 

25 

3^ 
37 
38 

9-30  33^ 
930  398 
9-30459 

62 
61 
6^ 

9-31  233 
9.31  297 
9-31  361 

0.68  767 
0.68  703 
0.68  639 

9-99  104 
9.99  lOI 
9-99  099 

3 
2 
3 
3 
2 

3 

24 

23 
22 

I 
2 
3 

4 
5 
6 

7 

5-9  0.3 
1 1.8  0.6 
17.7  0.9 
23.6  1.2 

29-5  1-5 
35-4  1.8 
41-3  2.1 

39 
40 

41 

9.30521 

61 
61 
61 

931  425 

0.68  575 

9.99  096 

21 
20 

19 

9.30  582 

9.31  489 

0.68  511 

9-99  093 

9-30  643 

9-31  552 

0,68  448 

9-99091 

42 

9.30  704 

61 

9.31  616 

0.68  384 

9.99  088 

18 

8 

47.2  2.4 

43 

9.30  765 

61 

9.31  679 

0.68  321 

9.99  086 

3 
3 

n 

9 

53-1  2.7 

44 
45 

9.30  826 
9.30887 

61 

60 

9-31  743 
9.31  806 

0.68  257 
0.68  194 

9.99  083 
9.99  080 

16 

IS 

4b 

9-30  947 

61 

9.31  870 

0.68  130 

9.99078 

3 
3 

14 

3   3   3 

47 

9.31  008 

60 

9-31  933 

0.68  067 

999075 

13 

67   66   65 

48 

9.31  068 

61 

9.31  996 

0.68  004 

9.99072 

12 

0 

49 
50 

51 

9.31  129 

60 
61 
60 

9-32059 

0.67  941 

9.99  070 

3 
3 
2 
3 
3 

II 
10 

9 

2 
3 

1 1.2  ii.o  10.8 

33-5  33-0  325 
55.8  55.0  54.2 

9.31  189 

9.32122 

0.67  878 

9.99  067 

9.31  250 

9.32  185 

0.67815 

9.99  064 

52 

9.31  310 

60 

9.32  248 

0.67  752 

9.99  062 

8 

53 

931  370 

60 

9.32  311 

0.67  689 

9-99  059 

7 

3   3   3 

,  54 

931  430 

60 

9-32  373 

63 
62 

63 
62 
62 

0.67  627 

9-99  056 

6 

64   63   62 

55 

9.31  490 

59 
60 
60 

59 
60 

932436 

0.67  564 

9-99  054 

5 

0 

5t> 

931  549 

9-32498 

0.67  502 

9.99051 

3 

4 

I 

10.7  10.5  10.3 

57 

9.31  609 

9.32561 

0.67  439. 

9.99  048 

J 

3 

2 

32.0  31.5  31.0 
53-3  52.5  51-7 

58 

9.31  669 

9-32  623 

0.67  377 

9-99  046 

3 
3 

2 

3 

59 
60 

9.31  728 

9-32685 

62 

0.6731? 

9  99  043 

I 
0 

9.31  788 

9-32  747 

0.67  253 

9-99  040 

L.  Cos. 

d. 

L.  Cot. 

c.  d.|  L.Tan.  | 

L.  Sin. 

d. 

f 

P.P. 

•7QC 


38 


/ 

L.  Sin.  1  d. 

L.  Tan. 

c.  d. 

L.  Cot. 

L.  Cos. 

d. 

P.  P. 

0 

9.3 1  788 

59 
60 

9-32  747 

63 
62 

0.67  253 

9.99  040 

2 

3 

3 
2 

60 

S9 

931  847 

9.32  810 

0.67  190 

9-99  038 

2 

9.31  907 

59 
59 
59 
59 
59 

9-32  872 

61 

0.67  128 

9-99  035 

58 

63  62  61 

3 

9.31  966 

932933 

6-' 

0,67  067 

9-99032 

57 

I 

6.3  6.2  6.1 

4 

9.32025 

932995 

6'> 

0.67  005 

9.99  030 

3 
3 

56 

2 

12.6  124  12.2 

5 

9.32084 

9-33057 

'  62 

0.66  943 

9.99027 

55 

3 

18.9  18.6  18.3 

6 

9.32  143 

9-33  "9 

61 

0.66  881 

9.99  024 

54 

4 

25.2  24.8  24.4 

7 

9.32  202 

9-33  180 

62 
61 

0.66  820 

9.99  022 

53 

5 

31.5  31.0  30.5 

8 

9.32261 

59 
58 
59 
59 
58 
58 
59 
58 
58 
58 
58 
58 
58 
58 

57 
58 
57 
58 
57 
57 
58 
57 
57 
57 
56 
57 
57 
57 
56 

9-33  242 

0.66  758 

9-99019 

3 
3 
3 
2 

3 
3 
3 

52 

6 

37-8  37-2  36-6 

9 
lO 

II 

932319 

9-33  303 

62 
61 
61 

0.66  697 

9.99016 

51 
50 

49 

I 

9 

44-1  43-4  42.7 
50.4  49.6  48.8 

56-7  55-8  54-9 

9.32  378 

9-33  365 

0.66  635 

9.99013 

9-32437 

933426 

0.66  574 

9.99  on 

12 

9-32495 

933487 

61 

0.66  513 

9.99  008 

48 

60   59 

13 

932553 

9-33  548 

61 

0.66452 

9-99  005 

47 

14 

9.32  612 

9.33  609 

6i 

0.66391 

9.99  002 

46 

I 

6.0   5.9 

15 

9.32  670 

9  33  670 

61 

0.66  330 

9.99  000 

3 

45 

2 

12.0  11.8 

lb 

9.32  728 

933731 

61 
61 
60 

0.66  269 

9.98  997 

44 

3 

18.0  17.7 

17 

9.32  786 

9-33  792 

0.66  208 

9.98  994 

3 

43 

4 

24.0  23.6 

18 

9.32  844 

933853 

0.66  147 

9.98991 

3 

2 

3 

3 

42 

5 
6 

7 
8 
9 

30-0  29.5 

19 
20 

21 

9.32  902 

9-33913 

61 
60 
61 

0.66087 

9-98  989 

41 
40 

39 

36.0  35.4 
42.0  41.3 
48.0  47.2 
54-0  53-1 

9.32960 

9-33  974 

0.66  026 

9.98  986 

9.33018 

9-34  034 

0.65  966 

9-98983 

22 

9-33075 

934095 

60 

0.65  905 

9.98  980 

:> 

38 

23 

9-33  ^33 

9-34155 

60 

0.65  845 

9.98978 

3 
3 
3 
2 

3 
3 
3 
3 
2 
3 
3 

37 

58   57 

24 

9-33  190 

9-34215 

61 

0.65  785 

9-98975 

36 

25 

9.33  248 

9-34  276 

60 

0  65  724 

9.98972 

35 

I 

5-8   5-7 

26 

9-33  305 

9-34336 

60 

0.65  664 

9.98  969 

34 

2 

1 1.6  1 1.4 

27 

933362 

9-34  396 

60 

0.65  604 

9.98  967 

33 

3 

17.4  17.1 
23.2  22.8 
29.0  28.5 
34-8  34.2 
40.6  39.9 
46.4  45.6 

28 

933420 

934456 

60 

0-65  544 

9.98  964 

32 

4 

29 
30 

31 

9-33  477 

9.34516 

60 

59 
60 

0.65  484 

9.98961 

31 
30 

29 

i 

9  33  534 

9-34  576 
9-34  635 

0.65  424 

9.98958 

933591 

0.65  365 

998955 

32 

9  33  647 

9-34  695 

60 

0.65  305 

9-98953 

28 

9 

52.2  51.3 

33 

9-33  704 

9-34  755 

59 
60 

0.65  245 

9  98  950 

27 

34 

9-33  761 

9-34814 

0.65  186 

9-98  947 

26 

56   55   3 

35 

9.33818 

934874 

59 

0.65  126 

9.98  944 

3 

25 

36 

9-33  874 

9  34  933 

0.65  067 

9.98941 

3 

24 

1 

50  5-5  0.3 

37 

9-33931 

56 

57 
56 
56 
56 
56 
56 
56 
55 
56 
55 
56 

55 
56 
55 
55 
55 

9.34992 

59 
59 
60 

0.65  008 

9.98938 

3 
0 

23 

2 
3 

4 

7 

II. 2  ii.o  0.6 
16.8  16.5  0.9 
22.4  220  1.2 
280  27.5  1.5 
33.6  33.0  1.8 
39.2  38.5  2.1 

3^ 

9-33987 

9-35051 

0.64  949 

9.98936 

3 
3 
3 
3 
3 

22 

39 
40 

41 

9-34  043 

9-35  III 

59 
59 
59 
59 
58 
59 
59 
58 
59 
58 
59 
58 
58 
58 
58 
58 
58 
58 
58 
58 
57 

0.64  889 

9-98  933 

21 
20 

19 

9.34  100 

9-35  170 

0.64  830 

9  98  930 

9.34  156 

9-35  229 

0.64  771 

9.98927 

42 

9.34212 

9-35  288 

0.64  712 

9  98  924 

18 

8 

44-8  44.0  2.4 

43 

9-34  268 

9-35  347 

0.64  653 

9.98  921 

17 

9 

50.4  49.5  2.7 

44 
45 

9-34  324 
9-34  380 

9-35  405 
9-35  464 

0-64  595 
0.64  530 

9.98919 
9.98  916 

3 

16 
15 

46 

9.34  436 

9-35  523 

0.64  477 

9.98913 

3 

14 

3   3   3 

47 

934491 

9  35  581 

0.64419 

9.98910 

3 

13 

62   61   60 

48 

9-34  547 

9  35  640 

0.64  360 

9.98  907 

J 

12 

0 

49 
50 

51 

9.34  602 

9.35  698 

0.64  302 

9.98  904 

3 
3 

II 
10 

9 

I 
2 
3 

10.3  10.2  lO.O 
31.0  30.5  30.0 
51.7  50.8  50.0 

9.34  658 

9-35  757 

0.64  243 

9.98901 

9-34713 

935815 

0.64  185 

9.98  898 

52 

9-34  769 

9-35  873 

0.64  127 

9.98  896 

8 

53 

9-34  824 

9-35  931 

0.64  069 

9-98893 

3 

7 

3   3   3 

54 

9-34879 

9-35  989 

0.64  01 1 

9.98  890 

3 
3 

6 

59   58   57 

55 

9-34  934 

9.36047 

0.63  953 

9.98  887 

5 

0 

5^ 

57 
58 

9-34  989 
9-35  044 
9.35  099" 

55 
55 
55 
55 
55 

9.36  105 
9.36  163 
9.36221 

0-63  895 
0.63837 
0.63  779 

9.98  884 
9.98881 
9.98  878 

3 
3 

3 
3 
3 

4 

3 
2 

I 
2 
3 

9.8  9.7  9.5 
29.5  29.0  28.5 
49.2  48.3  47.5 

59 
60 

9-35  154 

9.36  279 

0.63721 

9.98875 

I 
0 

9-35  209 

9-36  336 

0.63  664  9.98  872  1 

L.  Cos. 

d.   L.  Cot.  |c.  d.| 

'  L.Tan.  1  L.  Sin.  | 

d. 

P.P.      || 

WMO 


39 


L.  Sin. 


L.  Tan.  c.  d.  L.  Cot. 


L.  Cos. 


P.  P. 


O 

r 

2 

3 
4 

5 
6 

7 
8 

9 

10 

II 

12 

13 

15 
i6 

17 
i8 

19 

20 

21 
.22 
23 
24 

25 
26 

27 

28 

.29 

30 

31 
32 
33 
34 
35 
36 

37 
38 
39 
40 

41 

42 

43 
44 
45 
46 

47 
48 

49 

50 

51 
52 
53 
54 
55 
56 

57 
58 
59 
60 


9-35209 

9-35  263 
9-35318 
9-35  373 

9-35427 
9.35  481 

9-35  536 
9.35  590 
9-35  644 
9-35  ^98 


9-35  752 


9.35  806 
9-35  860 
9-35  9H 
9-35  968 
9.36022 
9.36075 

9.36  129 
9.36  182 
9-36  236 


9.36  289 


9-36  342 
9-36  395 
936449 
9.36  502 

9-36555 
9.36  608 

9  36  660 

9-36713 
9.36  766 


1.36  819 


9.36871 
9.36  924 
9.36  976 
9.37028 
9.37081 
9-37  ^33 
9-37  185 
9-37  237 
9-37  289 


9-37  341 


9-37  393 
9-37  445 
9-37  497 

9-37  549 
9.37  600 
9-37652 

9-37  703 
9-37  755 
9.37  806 


9.37  858 


9-37  909 
9-37960 
938011 
9.38062 

9-38113 
9.38  164 

9.38215 
9.38  266 
9-38317 


9.38  368 


9.36  336 


9  36  394 
9.36452 

9-36  509 
9.36  566 
9.36  624 
9.36681 

936  738 
9-36  795 
9.36852 


9.36  909 

9.36  966 
9.37023 

9.37  080 

9-37  ^37 
9-37  193 
9-37  250 
9-37  306 
9-37  363 
937419 


9-37  476 


9.37  532 
9-37  588 
9-37  644 
9.37  700 

9.37  756 
9-37812 

9.37  868 
9-37924 
9-37  980 


9-38  035 


9.38091 

9-38  147 
9.38  202 

9-38  257 
9-38313 
9.38  368 

9-38423 
9-38  479 
938  534 


9-38  5^ 


9.38  644 
9.38  699 
9-38  754 
9.38  808 
9.38  863 
9.38918 
9.38972 
9.39027 
9-39  082 


9-39  136 


9.39  190 

9-39  245 
9.39  299 

9-39  353 
9-39  407 
9.39461 

9-39515 
9-39  569 
9-39  623 


9-39  677 


0.63  664 
0.63  606 
0.63  548 
0.63491 
0.63  434 
0.63  376 
0.63319 
0.63  262 
0.63  205 
0.63  148 


9  98  872 
9.98  869" 
9.98  867 
9.98  864 
9.98861 
9.98  858 
9-98855 
9.98  852 
9.98  849 
9.98  846 


0.63091 
0.63  034 
0.62977 
0.62  920 
0.62  863 
0.62  807 
0.62  750 
0.62  694 
0.62  637 
0.62  581 


9.98  843 


9.98  840 
9.98837 
9.98  834 
9-98831 
9.98  828 
9.98  825 
9.98  822 
9.98819 
9.98816 


0.62  524 


9.98813 


0.62  468 
0.62  412 
0.62  356 
0.62  300 
0.62  244 
0.62  188 
0.62  132 
0.62  076 
0.62  020 


9.98  810 
9.98  807 
9.98  804 
9.98  801 
9.98  798 
9.98  795 
9.98  792 
9-98  789 
9.98  786 


0.61  965 


0.61  909 
0.61  853 
0.61  798 
0.61  743 
0.61  687 
0.61  632 
0.61  577 
0.61  521 
0.61  466 


9-98  783 
9.98  780 
9-98  777 
9.98  774 

9-98  77^ 
9.98  768 
9-98  765 
9.98  762 
9-98  759 
9-98  756 


0.61  411 


9-98  753 


0.61  356 
0.61  301 
0.61  246 
0.61  192 
0.61  137 
0.61  082 
0.61  028 
0.60973 
0.60918 


9-9^  750 
9.98  746 

9-98  743 
9.98  740 
998  737 
998  734 

998  731 
9.98  728 

9  98  725 


0.60  864 


9.98  722 


0.60810 
0.60  755 
0.60  701 
0.60  647 
0.60  593 
0.60  539 
0.60  485 
0.60431 
0.60  377 


9.98719 
9.98715 
9.98712 
9.98  709 
9.98  706 
9.98  703 
9.98  700 
9.98  697 
9  98  694 


0.60  323 


9.98  690 


60 

59 
58 
57 
56 
55 
54 

53 
52 
51 
50 

49 
48 
47 
46 
45 
44 

43 
42 
41 
40 

39 
38 
37 
36 
35 
34 

33 
32 
31 
30 

29 
28 
27 
26 

25 
24 

23 
22 
21 

20 

19 
18 

17 
16 
15 
14 

13 
12 
n 
10 

9 
8 

7 
6 
5 
4 

3 
2 
I  ■ 

O 


58  57   56 


5.8 
11.6 


5.7  5-6 
1.4  II. 2 
17.4  17.1  16.8 
23.2  22.8  22.4 
290  28.5  28.0 
34.8  34.2  33.6 
40.6  39.9  39.2 
46.4  45.6  44.8 
52.2  51.3  50.4 


55 

5-5 


54 

5.4 


53 


5-3 
no  10.8  10.6 
16.5  16.2  15.9 
22.0  21.6  21.2 
27.5  27.0  26.5 
33.0  32.4  31.8 

38.5  37-8  37.1 
44.0  43.2  42.4 
49.5  48.6  47.7 

52   51 


5-2 
10.4 
15.6 
20.8 
26.0 
31.2 

36.4 
41.6 
46.8 


0.4 
0.8 
1.2 
1.6 
2.0 
2.4 
2.8 

3-2 

3-6 


5.1 
10.2 

15-3 
20.4 

255 
30.6 

35-7 
40.8 

45.9 

3 

0.3 
0.6 
0.9 
1.2 

1-5 
1.8 
2.1 

2.4 

2-7 


55 


4 
54 


58 


6.9  6.8  9.7  9.5 
20.6  20.2  29.0  28.5 
34.433.848.347.5 
48.147.2  —  — 


56 


55 


54 


9.3  9.2  9.0 
28.0  27.5  27.0 
46.7  45-8  45  o 


L.  Cos. 


L.  Cot.  c.  d 


L.  Tan. 

7A° 


L»  Sin. 


P.  P. 


40 


!    '    1   L.  Sin. 

|d. 

L.  Tan.   |c.  d.     L.  Cot. 

L.  Cos. 

d. 

1               P.P. 

■ 

0 

9.38  368 

50 
51 
50 
51 

50 
50 

9-39677 

1 
54 

54 
53 

0.60  323 

9.98  690 

3 

60 

54        63 

9.38418 

9.39731 

0.60  269 

9.98  687 

2 

9.38  469 

9-39  78I 

0.60  2 1  5 

9.98  684 

3 

58 

3 

9.38519 

9-39  838 

0.60  162 

9.98  681 

3 

57 

I 

54      5-3 

4 

9.38  570 

9-39  892 

54 
53 

0.60  108 

9.98  678 

3 

S6 

2 

10.8     10.6 

5 

9.38  620 

9-39  945 

0.60055 

9.98  675 

3 

S5 

3 

16.2     15.9 

6 

9.38  670 

9.39  999 

54 

0.60001 

9.98  671 

4 

54 

4 

21.6     21.2 

7 

9.38721 

51 
50 
50 
50 
50 
50 
50 
50 
50 
49 

9.40  052 

53 
54 
53 
53 
54 
53 
53 
53 
53 
53 

0.59  948 

9.98  668 

3 

5S 

5 
6 

27.0    26.5 
324     31.8 
37.8     37-1 
43-2    424 
48.6     d.7.7 

8 

9-38  771 

9.40  106 

0.59  894 

9.98  665 

3 

52 

7 
8 
9 

9 
lO 

II 

9.38821 

9.40  159 

0.59  841 

9.98  662 

3 
3 
3 

51 
50 

49 

9.38871 

9.40  212 

0.59  788 

9.98  659 

9.38921 

9.40  266 

0.59  734 

9.98  656 

12 
13 

9-38971 
9.39021 

9.40319 
9.40  372 

0.59681 
0.59  628 

9.98  652 
9.98  649 

4 
3 

48 
47 

52      51      50 

14 

9.39071 

9.40  425 

0-59  575 

9.98  646 

3 

46 

I 

5.2    5-1     50 

15 

9.39  121 

9.40  478 

0.59  522 

9.98  643 

3 

45 

2 

10.4  10.2  lO.O 

16 

9.39170 

940531 

0.59  469 

9.98  640 

3 

44 

3 

15.6  15.3  15.0 

17 

9.39  220 

50 
50 
49 
50 
49 
49 
50 
49 
49 
49 

9.40  584 

53 
52 
53 
53 
53 
52 
53 
52 
53 
52 

0.59416 

9.98  636 

4 

A^ 

4 

20.8  20.4  20.0 
26.0  25.5  25.0 
31.2  30.6  30.0 

364  35.7  35.0 
41.6  40.8  40.0 
46.8  4';.q  4s.O 

18 

9.39  270 

9.40  636 

0.59  364 

9-98  633 

J 

42 

5 
6 

7 
8 

9 

19 

20 

21 

9.39319 

9.40  689 

0.59  31 1 

9.98  630 

3 
3 
4 
3 
3 
3 

41 
40 

39 

9.39  369 

9.40  742 

0.59  258 

9.98  627 

9.39418 

940  795 

0.59  205 

9.98623 

22 

9-39  467 

9.40  847 

0.59153 

9.98  620 

38 

23 

9.39517 

9.40  900 

0.59  100 

9.98617 

37 

49     48     47 

24 

9.39  566 

9.40952 

0.59  048 

9.98  614 

36 

25 

9.39  615 

9.41  005 

0.58  995 

9.98  610 

4 

35 

I  i 

4.9    4.0    4-7 

26 

9.39  664 

9.41  057 

0-58  943 

9.98  607 

3 

34 

2 

9.8     9.6     94 

27 

9-39713 

49 
49 

9.41  109 

52 
52 

0.58891 

q.98  604 

3 

^^ 

3 
4 

14.7   144  14.1 
19.6  19.2   18.8 

28 

9.39  762 

9.41  161 

0.58  839 

9.98601 

3 

32 

29 
30 

9-39  81 1 

49 
49 
49 
49 
48 
49 
48 
49 
48 

49 
48 

49 
48 
48 
48 
48 

9.41  214 

53 
52 
52 
52 
52 
52 
52 
52 
51 
52 
52 
51 
52 

51 
52 
51 

0.58  786 

9.98  597 

4 
3 
3 

31 

30 

I 

7 

24.5  24.0  23.5 
29.4  28.8  28.2 
34.3  33.6  32.9 

9.39  860 

9.41  266 

0.58  734 

9-98  594 

31 

9.39  909 

9.41  318 

0.58682 

9.98  591 

29 

8 

39.2  38.4  37.6 

32 

9.39  958 

9.41  370 

0.58  630 

9.98  588 

3 

28 

9 

44.1  43.2  42.3 

33 

9.40  006 

9.41  422 

0.58578 

9.98  584 

\ 

27 

34 

940055 

9.41  474 

0.58  526 

9.98  581 

3 

26 

4         3 

35 

9.40  103 

9.41  526 

0.58  474 

9.98578 

3 

25 

J 

0.4       0.3 
0.8       0.6 

3t> 

9.40152 

941  578 

0.58422 

9.98  574 

4 

24 

2 

^J 

9.40  200 

9.41  629 

0.58  21^ 

9.98568^ 

3 

3 
4 
3 

^ 

J 

1.2       0.9 

3^ 

9.40  249 

9.41  681 

0.58319 

22 

4 

1.6       1.2 

39 
40 

41 

9.40  297 

941  733 

0.58  267 

9.98565 

21 
20 

19 

I 

2.0       1.5 
2.4       1.8 
2.8       2.1 

9.40  346 

9.41  784 

0.58216 

9.98561 

9.40  394 

9.41  836 

0.58  164 

998558 

42 

9.40  442 

9.41  887 

0.58  113 

998555 

3 

18 

3.2       2.4 
3-6       2.7 

43 

9.40  490 

941  939 

0.58061 

9.98551 

4 

17 

9 

44 

9.40  538 

9.41  990 

0.58010 
0.57959 

9.98  548 
9.98  545 

3 

t6 

45 

9.40  586 

48 
48 
48 
48 
48 
47 
48 
48 

47 
48 

47 
48 

47 
47 

t8 

9.42  041 

51 
52 
51 
51 
51 
51 
51 
51 
51 
51 
51 
51 
50 
51 
51 

eri    1 

3 

15 

40 

9.40  634 

9.42  093 

0.57907 

9.98  541 

4 

14 

4444 

47 

9.40  682 

9.42  144 

0.57  856 

998  538 

3 

13 

54     53     52     51 

48 

9.40  730 

9.42  195 

0.57805 

998  535 

3 
4 
3 
3 

4 

12 

°    6.8    6.6    6.5    6.4 
2  20.2  19.9  19.5  19.1 

.33-8  33-1  325  31.9 
^47.246.445.544.6 

49 
50 

51 

9.40  778 

9.42  246 

0-57  754 

9.98  531 

II 
10 

9 

9.40  825 

9.42  297 

0.57  703 

9.98  528 

9.40  873 

9.42  348 

0.57652 

998  525 

52 

9.40921 

942  399 

0.57  601 

9.98521 

8 

53 
54 

9.409685 
9.41  016^ 

9.42  450 
9.42  501 

0.57  550 
0.57499 

9.98518 
9-98515 

3 
3 

7 
6 

A   A   A   1 

55 

9.41  063 

9-42  552 

0.57  448 

9.98  511 

4 
3 
3 
4 
3 
4 

5 

54    53     52     51 

5^ 

9.41  III 

9.42  603 

0.57397 

9.98  508 

4 

°   9.0    8.8    8.7    8.5 

57 

9.41  158 

942  653 

0-57  347 

9-98  505 

3 

27.0  26.5  26.0  25.5 

5^ 

9.41  205 

9.42  704 

0.57296 

9.98  501 

2 

^45.044.243.342.5 

59 

9.41  252 

942  755 

o-57'245 

9.98498 

I 

3 

60 

9.41  300 

9.42805  1  "     1  0.57195  1 

9.98  494 

0 

: 

L.  Cos. 

d.  1    L.  Cot.   |c.  d.     L.  Tan.  1    L.  Sin.    | 

d.l 

'1 

P.P. 

'y/;° 


41 


/ 

L.  Sin'. 

d. 

L.  Tan. 

c.  d. 

L.  Cot. 

L.  Cos. 

d. 

P.P. 

9.41  300 

47 

9.42  805 

5« 

50 

0.57  195 

9.98  494 

3 
3 

4 
3 
4 
3 
3 
4 
3 
4 
3 
4 
3 
3 
4 
3 
4 
3 
4 
3 
4 
3 
4 
3 
3 
4 
3 
4 
3 
4 
3 
4 
3 
4 
4 
3 
4 

I 

3 

4 
3 
4 
3 

60 

59 

61      50     49 

941  347 

9.42  856 

0.57  144 

9.98491 

1       2 

9.4'  394 

47 

9.42  906 

0.57094 

9.98  488 

58 

3 

9.41  441 

47 

942  957 

5^ 

0.57  043 

9.98  484 

57 

^ 

5-»     5-0    4-9 

10.2  lo.o    9.8 

15.3  15.0  14.7 

20.4  20.0  19.6 

25.5  25.0  24.5 

30.6  30.0  29.4 

i     4 

9.41488 

4/ 

943  007 

S^ 

0.56  993 

9.98481 

56 

2 
3 
4 

'     ,S 

9-41  535 

47 
47 
46 

47 

46 

47 
46 

943057 

50 
5» 
50 
50 
50 
50 
50 
50 
50 
50 
50 
49 
50 
50 
49 
50 
49 
50 
49 
50 
49 
49 
49 
50 
49 
49 
49 

0.56  943 

9.98477 

55 

i     6 

!    7 

9.41  582 
9.41  628 

943  loS 
943  158 

0.56  892 
0.56  842 

9.98  474 
9.98471 

54 
53 

1    ^ 

9.41  675 

943  208 

0.56  792 

9.98  467 

52 

7 

35-7  35-0  34-3 

1     9 

1  10 

II 

9.41  722 

943  258 

0.56  742 

9.98  464 

51 
30 

49 

8 
9 

40.8  40.0  39.2 

45.9  45.0  44.1 

9.41  7^« 
9.41  815 

943  308 

0.5b  b92 

9.98  460 

943  358 

0.56  642 

9.98457 

12 

9.41  8b  I 

943  408 

0.56  592 

9-98  453 

48 

48     47      46 

13 

15 

9.41  908 

941  954 
9.42001 

47 
46 

47 
46 

46 

^I 
46 

46 

46 

46 

46 

46 

45 
46 

46 

46 

45 
46 

45 
46 

45 
46 

45 
45 
46 

45 
45 
45 
45 
45 
45 
45 

943  458 
943  508 
943  558 

0.56  542 
0.56492 
0.56442 

9.98  450 
9.98  447 
9.98  443 

47 
46 
45 

2 

3 
4 

4.8    4.7    4.6 

9.6    9.4    9.2 

14.4  14.1   13.8 

19.2  18.8  18.4 

lb 

9.42  047 

943  607 

0.56  393 

9.98  440 

44 

17 
i8 

9.42  093 
9.42  140 

943  657 
943  707 

056  343 
0.56  293 

9.98  436 
9-98433 

43 
42 

24.0  23.5  23.0 
28.8  28.2  27.6 

19 
20 

21 

9.42  186 
942  232 

943  756 

0.56  244 

9.98  429 

41 
40 

39 

7 
8 

9 

33.6  32.9  32.2 
38.4  37.6  36.8 
43.2  42.3  41.4 

943  80b 

0.56  194 

9.98  426 

9.42278 

943  855 

0.5b  145 

9.98422 

22 

9.42  324 

943  905 

0.56095 

9.98419 

38 

1 

23 

9.42  370 

943  954 

0.56  046 

9.98415 

37 

45        44        1 

24 

25 

9.42416 
9.42  461 

9.44  004 
944053 

0.55  996 
0-55  947 

9.98412 
9.98  409 

36 

35 

2 

4-5       4.4 
9.0      8.8 

2b 

942  507 

9.44  102 

0.55  898 

9.98  405 

34 

3 

13.5     13-2 

27 

942  553 

9.44151 

0.55  849 

9.98  402 

33 

4 

18.0     17.6 

28 

942  599 

9.44  201 

0.55  799 

9.98  398 

32 

5 

22.5     22.0 

29 

30 

31 

9.42  b44 

944  250 

0.55  750 

9-98  395 

31 
30 

29 

I 

27.0     26.4 
31.5     30.8 
36.0    35-2 
40.5     39-6 

9.42  690 

944  299 

0-55  701 

9.98391 

942  735 

944  348 

0.55652 

9.98  388 

32 

9.42  781 

944  397 

49 
49 
49 
49 
48 
49 

0.55  603 

9.98  384 

28 

9 

33 

9.42  826 

944  446 

0.55  554 

9.98381 

27 

34 

9.42  872 

944  495 

0.55  505 

9.98  377 

26 

4          3 

3S 

9.42917 

944  544 

0.55  456 

9-98  373 

25 

I 

0.4      0.3 

36 

9.42  962 

944  592 

0.55  408 

9.98  370 

24 

2 

0.8      0.6 

37 

943  008 

944  641 

0.55  359 

9.98  366 

2^ 

' 

1.2      0.9 

3« 

943  053 

944  690 

49 
48 
49 
49 
48 
49 
48 

0.55310 

998  363 

22 

4 

1.6       1.2 

39 
40 

41 

943  09^ 

944  738 

0.55  262 

9.98  359 

21 
20 

19 

1 

9 

2.0       1.5 
2.4       1.8 
2.8       2.1 
3.2       2.4 
3.6       2.7 

943  143 

944  787 

0.55213 

998356 

943  i«« 

944  836 

0.55  164 

9.98352 

42 
43 
44 
4S 

943  233 
943  27^ 
943  323 
943  367 

9.44  884 
9-44  933 
9.44981 
945  029 

0.55  116 

0.55  067 

0.55019 
0.54971 

9  98  349 
9-98  345 
9.98  342 
9.98  338 

18 
17 
16 

44 
45 

48 

4 

15 

46 

9.43412 

945  078 

49 
48 
48 
48 
49 
48 
48 
48 
48 
48 
48 
47 
48 
48 
48 

0.54922 

9.98  334, 

4 

H 

4      4      4      4 

47 

943  457 

45 
45 
44 
45 
44 
45 
44 
45 
44 
44 
44 
45 
44 
44 

945  126 

0.54  874 

998331 

3 

4 

13 

50    49    48    47 

48 

943  502 

945  174 

0.54  826 

9.98327 

12 

0 

6.2   6.1    6.0   5.9 
18.8,18.418.017.6 
31.2  30.6  30.0  29.4 
43.842.942.041.1 

49 
50 

SI 

943  546 

945  222 

0.54  778 

9.98  324 

4 
3 

II 
10 

9 

I 
2 
3 

943  591 

945  271 

0.54  729 

9.98  320 

943  635 

945  319 

0.54681 

9.98317 

52 

9.43  680 

945  367 

0.54  633 

9.98313 

4 
4 
3 
4 
3 
4 
4 

8 

4                                1 

53 

943  724 

945415 

0.54  585 

9.98  309 

7 

3       3       3       3 
51     50    49    48 

54 

55 

943  769 
943813 

945  463 
945  511 

0.54537 
0.54489 

9.98  306 
9.98  302 

6 

5 

5t> 

943857 

945  559 

0.54441 

9.98  299 

4 

0 

8.5    8.3   8.2    8.0 

57 

943  901 

945  606 

0.54  394 

9.98  295 

3 

25.5  25.0  24.5  24.0 

5« 

943  94t> 

945  654 

0.54  346 

9.98  291 

2 

42.541.740.840.0 

59 
60 

943  990 

945  702 

0.54  298 

9.98  288 

3 
4 

0 

3 

944  034 

945  750 

0.54  250 

9.98  284 

L.  Cos. 

d. 

L.  Cot. 

c.  d. 

L.  Tan. 

L.  Sin. 

d. 

/ 

P.P. 

42 


w 


L.  Sin. 


L.  Tan.   Ic.  d.l    L.  Cot. 


L.  Cos. 


P.  P. 


7 
8 

9 

10 

II 

12 

13 
H 
'5 
i6 

17 
i8 

19 

20 

21 
22 
23 

24 

25 
26 

27 
28 
29 

30 


33 
34 
35 
36 

37 
38 
39 
40 

41 
42 

43 
44 
45 
46 

47 
48 
49 
50 

51 
52 
53 
54 
55 
56 

57 
58 
59 

60 


9-44  034 
9.44  078 
9.44  122 
9.44  166 
9.44  210 

9-44  253 
9.44  297 

9-44  341^ 
9-44  385 
9-44  428 


9.44  472 


9.44510 

9-44  559 
9.44  602 

9.44  646 
9.44  689 
9-44  733 
9.44  776 
9.44819 
9.44  862 


9-44  905 
9.44  948 

9.44  992 
9-45  035 

945077 

9.45  120 

9-45  '63 
9  45  206 
9-45  249 
9-45  292 


945334 


945  377 
945  4"  9 
945  462 
945  504 
945  547 
945  589 
945  632 
945  ^74 
945716 


945  758 


945  801 
945  843 
945  885 
945  927 
945  969, 
9,46011 

946053 
9.46095 
9.46  136 


..^^46,178 


9.46  220 
9.46  262 
946  3<^3 

946  345 
9.46  386 
9.46428 
9.46  469 
9.46  511 
946  552 


946  594 


44 
44 
44 
44 
43 
44 
44 
44 
43 
44 
44 
43 
43 
44 
43 
44 
43 
43 
43 
43 
43 
44 
43 
42 

43 
43 
43 
43 
43 
42 

43 
42 
43 
42 

43 
42 

43 
42 
42 
42 

43 
42 
42 
42 
42 
42 
42 
42 
41 
42 

-42 
42 
41 
42 

41 
42 
41 
42 
41 
42 


945  750 


945  797 
945  845 
945  892 
945  940 
945  987 
9.46  035 

9.46082 
9.46  130 

9.46  177 


9.46  224 
9.46  271 
9.46319 
9.46  366 
9.46413 
9.46  460 
9.46  507 

946  554 
9.46  601 
9.46  648 


946  694 
9.46741 
9.46  788 

946  835 
9.46881 

9.46  928 
946975 
9.47021 

9.47  068 

947  »M 


9.47  160 
947  207 
947  253 
947  299 
947  346 
947  392 
947  438 
947  484 
947  530 
947  576 


1.47622 


9.47  668 
9.47714 
947  760 
9.47  806 
9.47852 
947  897 
947  943 

947  989 

948  035 
9.48080 


9.48  126 
9.48  171 
9,48217 
9.48  262 
9.48  307 
948353 
9.48  398 

948  443 
9.48  489 


948  534 


47 
48 
47 
48 

47 
48 

47 
48 
47 
47 
47 
48 
47 
47 
47 
47 
47 
47 
47 
46 

47 
47 
47 
46 
47 
47 
46 

47 
46 
46 

47 
46 
46 
47 
46 
46 

46 
46 
46 
46 
46 
46 
46 
46 
46 
45 
46 
46 
46 

45 
46 

45 
46 

45 
45 
46 

45 

45 
46 

45 


o>54  250 


9.98  284 


0.54  203 

0-54155 
0.54  108 

0.54  060 
0.54013 
0.53  965 
0.53.918 
0.53  870 
0-53  823 


9.98281 
9.98  277 
9.98  273 
9.98  270 
9.98  266 
9.98  262 
9.98  259 
9-98  255 
9.98251 


0.53  776 


9.98  248 


0.53  729 
0.53  681 

0.53  634 
0-53  587 
0.53  540 
0.53  493 
0.53  446 
0.53  399 
053352 


9.98  244 
9.98  240 
9.98  237 

9-98  233 
9.98  229 
9.98  226 
9.98  222 
9.98218 
9.98215 


0.53  306 


0-53  259 
0.53  212 
0.53  165 
0.53  119 
0.53072 
0.53025 
0.52979 
0.52932 
0.52886 


9.98  21 1 
9.98  207 
9.98  204 
9.98  200 
9.98  196 
9.98  192 
9.98  189 
9.98  185 
9.98  181 
9.98177 


0.52  840 

0.52  793 
0.52  747 
0.52  701 
0.52  654 
0.52  608 
0.52  562 
0.52  516 
0.52470 
0.52424 


9.98174 


9.98  170 
9.98  166 
9.98  162 
9.98  159 
998  155 
9-98151 
9.98  147 
9.98  144 
9.98  140 


0.52378 


9.98  136 


0.52  332 
0.52  286 
0.52  240 
0.52  194 
0.52  148 
0.52  103 
0.52057 
0.52011 
0.51965 


9.98  132 
9.98  129 


125 
121 
117 
113 
no 
106 


9.98  102 


0.51  920 


0.51  874 
0.51  829 
0.51  783 

0.51  738 
0.51  693 
0.51  647 
0.51  602 

0-51  557 
0.51  511 


9.98  098 
^^^094" 
9.98  090 
9.98087 

9.98  083 
9.98  079 
9.98075 
9.98071 
9.98  067 
9  98  063 


0.51  466 


9.98  060 


60 

59 
58 

57 
56 

55 
54 

53 
52 
5» 
50 

49 
48 
47 
46 
45 
44 

43 
42 
41 
40 

39 
38 
37 
36 
35 
34 

33 
32 
31 
30 

29 
28 
27 
26 
25 
24 

23 
22 
21 

20 

19 
18 

'7. 
16 

15 
14 

13 
12 
II 
10 

9 
8 

7 
6 
5 
4 

3 
2 
I 

O 


48  47  46 


I 

4.8    4-7     4-6 

2 

9.6    9.4     9.2 

0 

144  H-i   13-8 

4 

1,9.2  18.8  18.4 

5 

24.0  23.5  .23.0 

6 

28.8  28.2  27.6 

7 

33.6  32.9  32.2 

8 

38.4  37.6  36.8 

9 

43.2  42.3  41.4 

45     44     43 

I 

4-5     44    4-3 

2 

9.0     8.8     8.6 

3 

13.5   13.2  12.9 

4 

18.0  17.6  17.2 

5 

22.5  22.0  21.5 

27.0  26.4  25.8 

31.5  30.8  30.1 
36.0  35.2  34.4 

40.5  39-6  38.7 
42   41 


4.2 
8.4 
12.6 
16.8 
21.0 
25.2 
29.4 
33-6 
37-8 


0.4 
0.8 
1.2 
1.6 
2.0 
24 
2.8 
3-2 
3.6 


4.1 
8.2 
12.3 
16.4 
20.5 
24.6 
28.7 
32.S 
36.9 


0.3 
0.6 
0.9 
1.2 

1-5 
1.8 
2.1 

2.4 

2.7 


4^ 
48 


4 

47 


46 


6.0  5.9  5.8  5.6 
18.0  17.6  17.2  16.9 
300  29.4  28.8  28.1 
42.0  41. 1  40.239.4 


3^ 
48 


3_ 

47 


46 


45 


8.0  7.8  7.7  7.5 
24.0  23.5  23.0  22.5 
40.0  39.2  38.3  37.5 


L.  Cos.  I  d. 


L.  Cot.  |c.  d.|  L.  Tan. 


L.  Sin. 


P.  P. 


43- 


L.  Sin. 


L.  Tan. 


c.  d< 


L.  Cot.   L.  Cos 


d. 


P.P. 


2 

3 

4 

5 

0 

7 
8 

9 

10 

II 

12 

13 

14 
J5 
i6 

17 
18 

19 

20 

21 

22 
23 
24 

25 
26 

27 
28 
29 

30 

31 

32 
33 
34 
35 
36 

Z1 
38 
39 
40 

41 
42 
43 
44 
45 
46 

47 
48 

49 

50 

51 
52 
53 
54 
55 
56 

57 
58 
59 
60 


46  594 


46633 
46  676 
46717 
46758 
46  800 
46  841 
46882 

46923 
46964 


4700I 


47045 
47086 
47127 
47  168 
47209 
47249 
47290 
47330 
47371 


47  41 1 


47452 
47492 

47  533 
47  573 
47613 
47654 
47694 
47  734 
47  774 


.47814 


•47  854 
47  894 
•47  934 

•47  974 
,48014 
,48  054 
.48094 
■48  "i^ZZ 
.48173 


48213 


48  252 
48  292 
48332 

48371 
48  41 1 

48  450 
48  490 
48  529 
48  568 


48  507 


48647 
48686 
48725 

48764 
48803 
48842 
48881 
48920 
48959 


48998 


948  534 


948579 
9.48  624 
9.48  669 
9.48714 
948  759 
9.48  804 

9  48  849 
9.48  894 
948  939 


9.48  984 


9.49  029 

949  073 
9.49  118 

949  163 
949  207 
949  252 
949  296 
949  341 
949  385 


949  430 


949  474 
949519 
949  563 
949  607 
949  652 
9.49  696 

949  740 
949  784 
9.49  828 


949  872 
9.49916 
949  960 
9.50  004 

9.50  048 
9.50092 
9.50  136 
9.50  180 
9.50  223 
9.50  267 


•50 


9.50355 
950398 
9.50  442 

9-50485 
9-50  529 
9-50572 
9.50616 
9-50659 
950703 


9-50  746 


9.50  789 

9-50  ^2,:^ 
9.50876 

9.50919 
9.50  962 
9.51005 
9.51 048 
9.51 092 
9-51 135 


9.51 178 


45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
44 
45 
45 
44 
45 
44 
45 
44 
45 
44 
45 
44 
44 
45 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
43 
44 
44 
44 
43 
44 
43 
44 
43 
44 
43 
44 
43 
43 
44 
43 
43 
43 
43 
43 
44 
43 
43 


0.51 466 
0.51 421 
0.51  376 

0.51  33» 

0.51  286 
0.51  241 
0.51  196 

0.51  151 
0.51  106 
0.51  061 


9.98  060 


9.98  056 
9.98052 
9.98  048 
9.98  044 
9.98  040 
9-98  036 
9.98  032 
9.98  029 
9.98023 


0.51  016 


9.98021 


0.50971 
0.50927 
0.50  882 
0.50  837 
0.50  793 
0.50  748 

0.50  704 
0.50  659 
0.50615 


9.98017 
9-98013 
9.98  009 
9.98005 
9.98001 
9-97  997 

9-97  993 
9.97  989 
9.97  986 


0-50  570 


9-97  982 


0.50  526 
0,50481 
0.50437 
0.50  393 
0.50  348 
0.50  304 
0.50  260 
0.50  216 
0.50  172 


9-97978 
9-97  974 
9.97  970 

9-97  966 
9-97  962 
9-97  958 
9-97  954 
9  97  950 
9.97  946 


0.50  128 
0.50  084 
0.50  040 
0.49  996 
0.49  952 
0.49  908 
0.49  864 
0.49  820 
0.49  777 
0-49  733 


9.97  942 


9-97  938 

9-97  934 

9-97  930 

9.97926 

9.97922 

9.97918 

9.97914 

9.97910 

9.97  906 

0.49  689 


9-97  902 


0.49  645 
0.49  602 
049  558 

0.495^5 
0.49471 
0.49  428 

0.49  384 
0.49  341 
0.49  297 


9.97  898 
9.97  894 
9-97  890 
9.97  886 
9.97  882 
9.97  878 
9-97  874 
9  97  870 
9.97  866 


0-49  254 


9.97  861 


0.49  2 1 1 
0.49  167 
0.49  124 
0.49081 
0.49  038 
0.48  995 
0.48952 
0.48  908 
0.48  865 


9-97857 
9-97  853 
9-97  849 

9.97  845 
9.97841 

9-97  837 

9-97  833 
9.97  829 
9.97  823 


0.48  822 


9.97  821 


60 

59 
58 
57 
56 
55 
54 

53 
52 
5^ 
50 

49 
48 

47 
46 
45 
44 

43 
42 
41 
40 

39 
38 
37 
36 
35 
34 

3?> 
32 
3^ 
30 

29 
28 
27 
26 
25 
24 

23 
22 
21 

20 

19 
18 

17 
16 

15 
14 

13 
12 
II 

10 

9 
8 

7 
6 

5 
4 

3 
2 

I 


45     44     43 


I 

4-5 

2 

9.0 

3 

U-5 

4 

18.0 

5 

22.5 

6 

27.0 

7 

3'S 

8 

36.0 

9 

40.5 

4-4 

8.8 

13.2 

17.6 


4-3 

8.6 

12.9 

17.2 


22.0  21.5 
26.4  25.8 
30.8  30.1 
35-2  344 
39-6  38.7 


42     41     40 


I 

4.2 

4.1 

4.0 

2 

8.4 

8.2 

8.0 

3 

12.6 

12.3 

12.0 

4 

16.8 

16.4 

16.0 

S 

21.0 

20.  s 

20.0 

6 

25.2 

24.6 

24.0 

7 

29.4 

28.7 

28.0 

8 

33-6 

32.8 

32.0 

9 

37.8 

36.9 

36.0 

39  5  4     3 

3.9  0.5  0.4  0.3 

7.8  i.o  0.8  0.6 

1 1.7  1.5  1.2  0.9 

15.6  2.0  1.6  1.2 

19.5  2.5  2.0  1.5 

234  3-0  2.4  1.8 

273  '3-5  2.8  2.1 

31.2  4.0  3.2  2.4 

35.1  4.5  3.6  2.7 


43 

4-3 
12.9 
21.5 
30.1 
387 


4^ 
43 


4^ 
45 


4 
44 


5.6  5-5 
16.9  16.5 
28.1  27.5 
394  "38-5 


3^ 

45 


3^ 

44 


54     7-5  7-3 

i6.i  22,5  22.0 

26.9  37-5  367 

37.6    —  — 


L.  Cos. 


L.  Cot.    c.  d.    L.  Tan 


L.  Sin. 


P.  P. 


70° 


44 


/ 

L.  Sin. 

d.  1  L.  Tan. 

c.  d.i    L.  Cot.   1   L.  Cos. 

d.l 

P.P. 

0 

I 

9.48  998 

39 

39 

39 

38 

39 

39 

3^ 

39 

39 

3ii 

39 

38 

38 

39 

38 

38 

39 

3^ 

^^ 
38 

38 
^8 

9.51  178 

43 
43 

0.48  822 

9.97  821 

4 

60 

59 

9-49  037 

9.51  221 

0.48  779 

9.97817 

2 

949  076 

9.51  264 

0.48  736 

9.97812 

5 

58 

3 

9-49  115 

9.51  306 

42 
43 

0.48  694 

9.97  808 

4 

57 

43     42     41 

4 

949  153 

9-51  349 

0.48  651 

9.97  804 

4 

56 

I 

4-3   4-2  4-1 

5 

9.49  192 

9.51  392 

43 

0.48  608 

9.97  800 

4 

55 

2 

8.6    8.4     8.2 

b 

9.49  231 

9-51  435 

43 

0.48  565 

9-97  796 

4 

54 

3 

12.9  12.6  12.3 

7 

949  269 

9-51478 

43 

0.48  522 

9.97  792 

4 

5^ 

4 

17.2  16.8  16.4 

8 

949  308 

9-51  520 

42 

0.48  480 

9-97  788 

4 

52 

5 
6 

21.5  21.0  20.5 
25.8  25.2  24.6 
30.1  29.4  28.7 
344  33-6  32.8 
^8.7  ^7.8  ^6.0 

9 
10 

949  347 

9-51563 

43 
43 
42 
43 
43 
42 

43 

42 

42 

43 
42 
43 
42 

42 
42 

43 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
41 
42 
42 
42 
42 
41 
42 

41 

42 

42 

41 
42 

0.48  437 

9-97  784 

4 
5 

51 
50 

949  385 

9.51  606 

0.48  394 

9.97  779 

II 

949  424 

9.51648 

0.48  352 

9-97  775 

4 
4 

49 

9 

12 

9.49  462 

9.51  691 

0.48  309 

9.97771 

48 

13 

949  500 

9-51  734 

0.48  266 

9-97  767 

4 

47 

H 

949  539 

9.51  776 

0.48  224 

9-97  763 

4 
4 
5 
4 

46 

^5 

949  577 

9.51  819 

0.48  181 

9-97  759 

45 

39     38     37 

i6 
17 

9.49615 
949  654 

9.51  861 
9.51903 

0.48  139 
0.48  097 

9-97  754 
9-97  750 

44 
43 

I 
2 

3.9     3.8     3.7 
7.8     7.6     7.4 

i8 
19 
20 

2l' 

949  692 
949  730 

9.51  946 
9.51988 

0.48  054 
0.48012 

9.97  746 
9.97  742 

4 

: 

42 
41 
40 

39 

3 
4 
5 
6 

II. 7  11.4  II. I 
15.6  15.2  14.8 
19.5   19-0  18.5 
23.4  22.8  22.2 

949  70^ 

9.52031 

0.47  969 

9-97  738 

9.49  806 

9.52073 

0.47  927 

9-97  734 

22 

949  844 

J'' 
38 
38 
38 
38 
38 
38 
38 
38 
37 
38 
38 
37 
38 
38 
37 
38 
37 
37 
38 
37 
37 
38 
37 
37 

9-52  115 

0.47  885 

9-97  729 

5 

38 

7 

27.3  26.6  25.9 

23 

9.49  882 

9-52157 

047  843 

9-97  725 

4 

37 

8 

31.2  30.4  29.6 

24 

9.49  920 

9.52  200 

0.47  800 

9.97  721 

4 

36 

9 

35.1  34.2  33.3 

2,S 

949  958 

9-52  242 

0.47  758 

9.97717 

4 

35 

26 

949  996 

9.52  284 

0.47  716 

9-97713 

4 
5 
4 
4 
4 
5 
4 
4 
4 
5 
4 
4 
4 
5 
4 
4 
4 
5 
4 
4 

34 

27 

9-50034 

9-52  326 

0.47  674 

9-97  708 

33 

36       5       4 

28 

9.50072 

9-52  368 

0.47  632 

9.97  704 

32 

29 

30 

31 

9.50  no 

9.52410 

0.47  590 

9-97  700 

31 
30 

29 

2 
3 

4 

3.0    0.5    0.4    1 
7.2    i.o    0.8 
10.8    1.5    1.2 
14.4    2.0    1.6 
18.0    2.5    2.0 
21.6    3.0    2.4 

9.50  148 

9.52452 

0.47  548 

9  97  69^ 

9.50  185 

9-52494 

0.47  506 

9-97  691 

32 

9.50223 

9-52536 

0.47  464 

9-97  687 

28 

33 

9.50  261 

9-52578 

0.47  422 

9.97  683 

27 

34 

9.50  298 

9.52  620 

0.47  380 

9.97  679 

26 

7 

25.2    3.5    2.8 

35 

9-50336 

9.52  661 

047  339 

9-97  674 

25 

8 

28.8    4.0    3.2 

3^ 

9-50  374 

9.52  703 

0.47  297 

9-97  670 

24 

9 

32.4    4.5    3.6 

37 

9.50411 

9-52  745 

0.47  255 

9.97  666 

23 

3« 
39 
40 

41 

9.50449 

9-52787 
9.52829 

0.47  213 
0.47  171 

9.97  662 
9-97657 

22 
21 

20 

19 

9.50486 

5        5        5 

9-50523 

9.52870 

0.47  130 

9-97  653 

'9-50561 

9.52912 

0.47  088 

9.97649 

42 

9-50  598 

9-52953 

0.47  047 

9.97  645 

18 

A9.       A9.       A^ 

43 

9.50  635 

9.52995 

0.47  005 

9.97  640 

17 

44 

9.50  673 

9-53037 

0.46  963 

9.97  636 

16 

4.3    4.2    4.1 

45 

9.50710 

9-53078 

0.46  922 

9.97  632 

15 

2 

12.9  12.6  12.3 

46 

9-50  747 

9.53  120 

0.46  880 

9.97  628 

4 

H 

21.5  21.0  20.5 

47 
48 

9.50  784 
9.50821 

3' J 
37 

11 

37 
37 
37 
36 
37 
37 
37 
37 
36 
37 

9-53161 
9-53  202 

41 
41 
42 

41 
42 
41 
41 
41 
42 
41 
41 
41 
41 
41 

0.46  839 
046  798 

9-97  623 
9.97619 

5 
4 
4 
5 
4 
4 
5 
4 
4 
5 
4 
4 
5 
4 

13 
12 

4 
5 

30.1  29.4  28.7 
38-7  37-^  36.9 

49 
50 

SI 

9.50  858 

9-53  244 

0.46  756 

9.97615 

II 
10 

9 

±    A    A 

9.50  896 

9-53  285 

0.46715 

9.97  610 

9.50933 

9-53  327 

0.46  673 

9.97  606 

52 

9.50970 

9-53  368 

0.46  632 

9.97  602 

8 

53 
54 

9.51007 
9.51043 

9-53  409 
9-53450 

0.46  591 
0.46  550 

9-97  597 
9-97  593 

7 
6 

0 

43      42      41 

54    5-2     5-1 

55 

9.51  080 

9-53492 

0.46  508 

9-97  589 

5 

^ 

16.1   15.8  15.4 

5^' 

9-51  "7 

9-53  533 

0,46  467 

9-97  584 

4 

3 
4 

26.9  26.2  25.6 

1  ^^7 

9.51  154 

9-53  574 

0.46  426 

9-97  580 

3 

37.6  36.8  35.9 

58 

9.51  191 

9-53615 

0.46  38^ 

9-97  576 

2 

59 
60 

9.51  227 

9.53  656 

0.46  344 

9-97571 

0 

9.51  264 

9-53697 

0.46  303 

9-97  567 

L.  Cos. 

d. 

L.  Cot. 

c.  d.l    L.  Tan.  | 

L.  Sin. 

d. 

/ 

p.p. 

7V 


45 


r*^ 

L.  SinTT  d. 

L.  Tan.  !c.  d. 

L.  Cot.   L.  Cos. 

d. 

P.P. 

0 

I 

9.51  264 

37 

37 
36 

9-53  697 
9.53  738 

41 

0.46  303 

9.97  567 

4 

5 
4 
4 

5 
4 

60 

59 

9.51  301 

0.46  262 

9.97  563 

2 

3 

9-5133^ 
9-51374 

9-53  779 
9.53  820 

4* 
41 
41 
41 
41 
41 

41 
40 

41 
.41 
40 
41 
41 
40 
41 
40 

41 
40 
41 
40 
41 

0.46  221 
0.46  180 

9.97  558 
9.97  554 

58 
57 

41  40  39 

4 

9.51  411 

37 

36 

9.53861 

0.46  139 

9.97  550 

56 

I 

4.1  4.0  3.9 

8.2  8.0  7.8 

5 

9-51447 

9-53  902 

0.46  098 

9-97  545 

55 

2 

0 
7 

9.51  484 
9.51  520 

37 
36 
37 

9.53  943 
9-53  984 

0.46057 
0.46016 

9.97541 
9.97  536 

54 
53 

3 
4 

5 

12.3  12.0  II. 7 

16.4  16.0  15.6 

20.5  20.0  19.5 

24.6  24.0  23.4 

28.7  28.0  27.3 

32.8  32.0  31.2 

36.9  36.0  35.1 

8 

9.51  557 

9.54025 

0.45  975 

9.97  532 

4 
5 
4 
4 
5 

52 

6 

9 
10 

II 

9.51  593 
9.51  629 

36 

37 

"•A 

9-54065 

0-45  935 

9-97  528 

5* 
50 

49 

I 

9 

9.54  »o6 

0.45  894 

9-97  523 

9.51  666 

9-54  147 

0.45  853 

9-97  5^9 

12 

9.51  702 

^6 

9-54  187. 

0.45  813 

9-97  5'5 

48 

13 

9.51  738 

36 
37 

9.54  228 

0.45  772 

9.97510 

47 

14 

9.51  774 

9.54  269 

0.45  731 

9-97  506 

4 
5 
4 

I 

46 

37   36   35 

15 

9.51  811 

9-54  309 

0.45  691 

9-97  501 

45 

lb 

9.51847 

36 
36 
36 
36 
36 
^6 

9.54  350 

0.45  650 

9-97  497 

44 

I 

3-7  3-6  3-5 

17 

9.51883 

9.54  390 

0.45  610 

9.97492 

43 

2 

7.4  7.2  7.0 

18 

9.51  919 

9-54431 

0.45  569 

9-97488 

4 

42 

3 

11. 1  10.8  10.5 

19 
20 

9.51  955 

9.54471 

0-45  529 

9.97  484 

4 

4 
5 
4 
5 
4 
4 
5 
4 
5 
4 

5 

41 
40 

4 

I 

14.8  14.4  14.0 
18.5  18.0  17.5 

9-51  991 

9.54512 

0.45  488 

9.97  479 

21 

9.52027 

9-54  552 

0.45  448 

9.97475 

39 

7 

25.9  25.2  24.5 

22 

9.52063 

36 
36 

^t 
36 

35 

36 

36 

36 

35 

:56 

9-54  593 

40 
40 

0.45  407 

9.97  470 

38 

8 

29.6  28  8  28.0 

23 

9.52099 

9-54  633 

0.45  367 

9.97  466 

37 

9 

33-3  32.4  31-5 

24 

9-52135 

954673 

41 
40 
40 
41 
40 
40 
40 

40 
40 
40 
40 
40 
40 

40 
40 
40 
40 
40 
39 

0.45  327 

9.97461 

3b 

25 

9.52  171 

9.54  7H 

0.45  286 

9-97  457 

35 

26 

9.52  207 

9-54  754 

0.45  246 

9-97  453 

34 

27 

9.52  242 

9-54  794 

0.45  206 

9.97  448 

33 

34  5   4 

28 

9.52  278 

9.54835 

0.45  165 

9-97  444 

32 

I 

3-4  0.5  0.4 

29 

30 

31 

9-52314 

9.54  875 

0.45  125 

9.97  439 

31 
30 

29 

2 
3 

4 

6.8  i.o  0.8 
10.2  1.5  1.2 
13.6  2.0  1.6 

9-52350 

9.54915 

045  085 

9.97  435 

9-52  385 

9-54  955 

0.45  04? 

9-97  430 

32 

9.52421 

0" 

35 
36 

35 
36 

35 
36 
35 
36 

3^ 

lb 

9-54  995 

0.45  005 

9-97426 

4 
5 
4 

I 

5 
4 
5 
4 
5 
4 
5 

28 

5 

17.0  2.5  2.0 

33 
34 
35 
36 

37 

9-52456 
9-52492 
9-52527 
9-52563 
9-52  598 

9-55035 
9.55075 
9-55  "5 
9.55155 
9.55  195 

0.44  965 
0.44  925 
0.44  885 
0.44  8-45 
0.44  805 

9.97421 

9.97417 
9.97412 

9-97  408 

9-97  403 

27 
26 
25 
24 

23 

6 

I 

9 

20.4  3.0  2.4 
23-8  3-5  2.8 
27.2  4.0  3.2 
30.6  4.5  3.6 

3« 
39 
40 

41 

42 

9.52634 
9.52  669 

9-55  235 
9-55  275 

0-44  765 
0.44  725 

9-97  399 
9-97  394 

22 
21 

20 

19 
18 

5   5   5 
41  40   39 

9-52  705 

9-55315 

0.44  685 

9-97  390 

9.52  740 
9-52  775 

9.55  355 
9-55  395 

0.44  645 
0.44  605 

9.97  385 
9.97  381 

43 

9-52811 

9-55  434 

0.44  566 

9.97  376 

17 

0 

44 

9.52846 

35 

35 

35 

9-55  474 

40 
40 
40 
39 

0.44  526 

9.97  372 

4 
5 
4 

s 

16 

4.1  4.0  3.9 
12.3- 12.0  II. 7 
20.5  20.0  19.5 
28.7  28.0  27.3 

45 
46 

9.52881 
9.52916 

9-55  5H 
9-55  554 

0.44  486 
0.44  446 

9-97  367 
9.97  363 

15 
H 

2 
3 

47 

9-52951 

35. 

35 

35 

36 

34 

35 

35 

35 

35 

35 

35 

34 

35 

9-55  593 

40 
40 
39 
40 

39 
40 
39 
40 
39 
40 
39 
39 
40 

0.44  407 

9.97  358 

5 
4 
5 
4 
5 

13 

4 

36.9  36.0  35.1 

48 

9-52986 

9-55  633 

0.44  367 

9.97  353 

12 

5 

49 
50 

51 

9.53021 

9-55  673 

0.44  327 

9.97  349 

11 
10 

9 

4   4   4 

9.53056 

9.55  712 

0.44  288 

9-97  344 

9.53092 

9.55  752 

0.44  248 

9.97  340 

52 

9.53126 

9-55  791 

0.44  209 

9.97  335 

8 

53 

9.53  161 

9-55  831 

0.44  169 

9-97  331 

4 

5 
4 
5 
5 

7 

41  40  39 

54 

9.53  196 

9-55  870 

0.44  130 

9.97  326 

6 

5.1  5.0  4.9 

55 

9.53  231' 

9-55910 

0.44  090 

9-97  322 

S 

15.4  15.0  14.6 

S^ 

9.53  266 

9.55  949 

0.44051 

9-97317 

4 

3 
4 

25.6  25.0  24.4 

5-7 

9.53  301 

9-55  989 

0.44011 

9.97312 

3 

35-9  35-0  34-1 

5« 

9.53  336 

9.56028 

0.43  972 

9-97  308 

4 

5 
4 

2 

59 
60 

9.53  370 

9-56  067 
9.56  107 

0.43  933 

9-97  303 

0 

9.53  405 

0.43  893 

9.97  299 

1  L.  Cos.  1  d.  1  L.  Cot.  |c.  d.|  L.  Tan.  |  L.  Sin. 

1  d. 

/ 

1      P.P. 

7n' 


46 


20' 


L.  Sin. 


L.  Tan.  c.  d 


L.  Cot. 


L.  Cos. 


P.P. 


9 
10 

II 

12 

13 

14 

15 
16 

17 

i8 
19 
20 

21 

22 
23 

24 
25 
26 

27 
28 
29 

30 

31 
32 
33 
34 
35 
30 
37 
3^^ 
39 
40 

41 
42 
43 
44 
45 
46 

47 
48 
49 
50 

51 

52 
53 
54 
55 
56 

57 
58 
59 
60 


9.53  405 


9-53  440 
9-53  475 
9-53  509 
9-53  544 
9-53578 
9-53613 

9-53  647 
9.53682 
953716 


9.53751 


9-53  785 
9.53819 

953854 
9-53  888 
953922 
9-53  957 

9-53  99_i 
9-54025 

9-54  059 


9-54093 


9.54127 
9-54  161 
9-54  195 
9.54  229 
9-54  263 
9-54  297 
9-54  33' 
9-54  365 
9.54  399 


9-54  433 


9.54  466 
9-54  500 
9-54  534 

9-54  567 
9.54601 

9-54  635 
9.54  668 
9.54  702 
9-54  735 


9-54  769 
9.54  802 
9.54  836 
9.54  869 

9-54  903 
9.54936 
9-54  969 
955003 
9-55  036 
9-55  069 


9-55  '02 


9-55  '36 
9-55  169 
9.55  202 

9-55  235 
9.55  268 

9-55  301 
9-55  334 
9-55  367 
9-55  400 


9-55  433 


9.56  107 


9.56  146 
9.56  185 
9.56  224 
9.56  264 

9-56  303 
9.56  342 

9-56381 
9.56  420 

9-56459 


9-56  498 


9-56537 
9-56576 
956615 

9-56  654 
9.56  693 
9-56  732 
9.56771 
9.56  810 
9.56  849 


9-56  887 


9.56926 

9-56  965 
9.57004 

9.57042 
9.57081 
9.57  120 

9-57  158 
9-57  »97 
9-57  235 


9-57  274 


9-57  3'2 
9-57351 
9-57  389 
9.57428 
9.57466 
9-57  504 

9-57  543 
9-57581 
9.57619 


9-57658 


9.57  696 
9-57  734 
9-57  772 
9.57810 

9-57  849 
9.57887 

9.57925 
9.57  963 
9.58001 


58039 


9.58077 
9.58  115 
9-58153 
9.58  191 
9-58  229 
9.58  267 

9-58  304 
958342 
9.58  380 


9.58418 


39 
39 
39 
40 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
38 

39 
39 
39 
38 
39 
39 
38 
39 
38 
39 
38 
•39 
38 
39 
38 
38 
39 
38 
38 
39 
38 
38 
38 
38 
39 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
37 
38 
38 


0.43  893 


9-97  299 


0.43  854 

9.97  294 

0.43815 

9.97  289 

0.43  776 

9-97  285 

0-43  736 

9.97  280 

0.43  697 

9  97  276 

0.43  658 

9.97  271 

0.43  619 

9.97  266 

0.43  580 

9.97  262 

0.43  541 

9-97  257 

0.43  502 

9-97  252 

0.43  463 

9.97  248 

0.43  424 

9.97  243 

0.43  385 

9.97  238 

0.43  346 

9-97  234 

0.43  307 

9.97  229 

0.43  268 

9.97  224 

0.43  229 

9.97  220 

0.43  190 

9.97215 

0.43  151 

9.97  210 

0.43  I '3 


9.97  206 


0.43  074 

0.43  035 
0.42  996 

0.42958 
042-919 
0.42  880 
0.42  842 
0.42  803 
0.42  765 


9.97  201 
9-97  196 
9-97  192 
9.97  187 
9.97  182 
9-97  178 

9-97  173 
9.97  168 

9-97  ^63 


0.42  726 


9-97  159 


0.42  688 
0.42  649 
0.42  611 
042572 
0.42  534 
0.42  496 

0.42  457 
0.42419 
0.42  381 


9-97  154 
9-97  149 
9  97  145 
9-97  140 
9-97  135 
9.97  130 

9.97  126 
9.97  121 
9.97  116 


0.42  342 


9  97" 


0.42  304 
0.42  266 
0.42  228 
0.42  190 
0.42  151 
0.42  113 
0.42  075 
0.42037 
0.41  999 


9.97  107 
9  97  102 
9.97  097 
9.97  092 
9-97  087 
9-97  083 
9.97  078 

997073 
9.97  068 


0.41  961 


9.97  063 


0.41  923 
0.41  885 
0.41  847 
0.41  809 
0.41  771 
0.41  733 
0.41  696 
0.41  658 
0.41  620 


9-97  059 
997054 
9.97  049 

9-97  044 
9-97  039 
9-97035 
9-97  030 
9-97  025 
9.97  020 


0.41  582  I  9.97015 
L.  Tan.  [    L.  Sin. 

fift° 


60 

59 
58 
57 
56 
55 
54 

53 
52 
51 
50 

49 
48 
47 
46 
45 
44 

43 
42 
41' 
40 

39 
3^ 
37 
36 
35 
34 

33 
32 
31 
30 

29 
28 
27 
26 
25 
24 

23 
22 
21 
20 

19 
18 

17 
16 
15 

14 

13 
12 
II 

10 

9 
8 

7 
6 
5 
4 

3 
2 


40     39 


4.0 
8.0 


12.0 
16.0 
20.0 


7.6 
II.4 
15.2 

19.0 


3-9 

78 

11.7 

15.6 

19-5 
24.0  23.4  22.8 
28.0  27.3  26.6 
32.0  31.2  30.4 
36.0  35.1  34.2 


37     35     34 


3-7 

7-4 

II. I 

14.8 
18.5 


3.4 

6.8 

10.2 

13.6 

17.0 


3.5 

7-0 
10.5 
14.0 

17-5 
22.2  21.0  20.4 
25.9  24.5  23.8 
29.6  28.0  27.2 
33-3  31.5  30.6 


33 


I 

3.3 

0.5 

0.4 

2 

6.6 

I.O 

0.8 

3 

9-9 

1.5 

1.2 

4 

13.2 

2.0 

1.6 

5 

16.S 

2.5 

2.0 

6 

19.8 

30 

2.4 

7 

23.1 

.3-5 

2.8 

8 

26.4 

4.0 

3-2 

9 

29.7 

4.5 

3-6 

AAA 

40      39      38 

4.0  3-9     3-8 

12.0  II. 7  11.4 

20.0  19.5   19.0 

28.0  27.3  26.6 

36.0  35.1  34.2 


AAA 

37      39      38 


3-7    4-9 
[I.I   146 


4-8 
[4.2 


18.5  24.4  23.« 
25-9  34-1  33.2 
33-3    —     — 


L.  Cos. 


d. 


L.  Cot. 


c.  d. 


P.  P. 


47 


O 

I 

2 

3 

4 

5 
6 

7 
8 

9 

10 

II 

12 
13 
14 
15 
16 

17 
18 

19 

20 

21 
22 
23 

24 
25 
26 

27 
28 
29 
30 

31 
32 
33 
34 
35 
36 

37 
38 
39 
40 

41 
42 
43 
44 
45 
46 

47 
48 

49 

50 

51 

52 
53 
54 
55 
56 

57 
58 
59 
60 


L.  Sin. 

9-55  43: 


9-55  4^^ 
9.55  499 
9.55  532 
9-55  564 
9-S5  597 
9-55  630 
9-55  663 
9-55  695 
9-55  728 


9-55  761 


9-55  793 
9.55  826 

9-55  858 
9-55891 
955923 
9-55  956 
9  55  988 
9.56021 

956053 


9-56  085 


9.56  118 
9.56  150 
9.56  182 
9.56215 
9.56  247 
9.56  279 

9563" 
9-56  343 
9.56375 


9.56  408 

9-56  440 
9.56472 
9.56  504 

9-56536 
9.56  568 
9-56  599 
9.56631 
9.56  663 
9-56  695 


9-56  727 


956  759 
9-56  790 
9.56822 

9.56854 
9.56  886 
9.56917 
9.56  949 
9.56  980 
9.57012 


9-57  044 


9-57075 
9.57  107 

9-57  138 
9.57  169 
9.57  201 
957232 
9.57  264 
9-57  295' 
9-57  326 


9-57  358 

L.  Cos.    I  d. 


L.  Tan.    c.  d.     L.  Cot. 


9.58418 

958455 
958493 
9.58531 
9.58  569 
9.58606 
9:58  644 
9.58681 
9.58719 
9-58757 


9.58  794 
9.58  832 
9-58  869 
9.58907 

958  944 
9-58981 
9.59019 
9.59056 
9-59  094 
9-59  131 


9.59  168 


9.59  205 
9-59  243 
9.59  280 

9.59317 
9-59  354 
9-59  391 

9-59  429 
9.59466 

9-59  503 


9-59  540 

9-59  577 
9.59614 

9.59651 
9.59  688 
9-59  725 
9-59  762 
9-59  799 
9-59  835 
9.59872 


9-59  909 


9-59  946 

9.59  983 
9.60019 

9.60056 

9.60  093 
9.60  130 
9.60  166 
9.60  203 
9.60  240 


9.60  276 


9.60313 
9.60349 
9.60  386 
9.60  422 
9.60459 
9-60  495 
9.60  532 
9.60  568 
9.60  605 


9.60  641 

L.  Cot. 


37 
38 
38 
38 
37 
3^ 
37 
38 
38 
37 
38 

37 
38 

37 
37 
38 
37 
38 
37 
37 
37 
38 
37 
37 
37 
37 
38 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
36 
37 
37 
37 
37 
36 

37 
37 
37 
36 
37 
37 
36 

37 
36 
37 
36 
37 
36 
37 
36 
37 
36 


Q.41  582 


0.41  545 
0.41  507 
0.41  469 
0.41  431 
0.41  394 
0.41  356 
0.41  319 
0.41  281 
0.41  243 


0.41  206 


0.41  168 
0.41  131 
0.41  093 
0.41  056 
0.41  019 
0.40  981 
0.40  944 
0.40  906 
0.40  869 


0.40  832 


0.40  795 
0.40  757 
0.40  720 
0.40  683 
0.40  646 
0.40  609 
0.40571 
0.40  534 
0.40  497 


0.40  460 


0.40  423 
0.40  386 
0.40  349 
0.40312 
0.40  275 
0,40  238 
0.40  201 
0.40  165 
0.40  1 28 


0.40091 


0.40  054 
0.40017 
0.39  981 
0.39  944 
0.39  907 
0.39  870 
0.39  834 
0.39  797 
0.39  760 


0.39  724 


0.39  687 
0.39651 
0.39614 

0.39  578 
0.39  541 
0.39  50? 
0.39  468 
039  432 
0.39  395 
0.39  359 
c.  d.|  L.  Tan. 


L.  Cos. 

9.97015 


9.97  010 
9-97  005 
9.97  001 

9.96  996 
9.96991 
9.96  986 
9.96981 
9.96  976 
9.96971 


9.96  966 


9.96  962 

9-96957 
9.96952 

9-96  947 
9.96  942 
9-96  937 
9-96  932 
9.96927 
9.96922 


9.96917 


9.96  912 
9-96  907 
9.96  903 

9.96  898 
9.96  893 
9.96  888 
9.96  883 
9.96  878 
996  873 


9.96  868 


9.96  863 
9.96  858 
996853 
9.96  848 
9.96  843 
9.96  838 
9.96  833 
9.96  828 
9.96823 


9.96818 


9.96813 
9.96  808 
9.96  803 
9.96  798 

996  793 
9.96  788 

9.96  783 
9.96  778 
9.96  772 


.96  767 


9.96  762 

996757 
9-96  752 
9.96  747 
9.96  742 
996  737 
9.96  732 
9.96  727 
9  96  722 

996  7^7 
L.  Sin. 


vr 


60 

59 
58 
57 
56 

55 
54 

53 
52 
51 
50 

49 
48 
47 
46 
45 
44 

43 
42 
41 
40 

39 
38 
37 
36 
35 
34 

33 
32 
31 
30 

29 
28 
27 
26 
25 
24 

23 
22 
21 

20 

19 
18 

17 
16 

15 
14 

13 
12 

10 

9 


P.  P. 


38  37 


3-8 

7.6 

11.4 

15.2 
190 


3.7  3-6 
7-4  7-2 
II. I  10.8 
14.8  14.4 
18.5  18.0 
22.8  22.2  21.6 
26.6  25.9  25.2 
30.4  29.6  28.8 
34.2  33-3   32.4 


33  32  31 


3-3 
6.6 

9-9 
13.2 
16.5 
19.8 
23.1 


3.2 

6.4 

9.6 

12.8 

16.0 

19.2 

22.4  21.7 
26.4  25.6  24.8 
29.7  28.8  27.9 


9-3 
12.4 

J5-5 
18.6 


I 

0.6 

0-5 

0.4 

2 

1.2 

I.O 

0.8 

3 

1.8 

1-5 

1.2 

4 

2.4 

2.0 

1.6 

5 

30 

2-5 

2.0 

6 

3-6 

30 

2.4 

7 

4.2 

3-5 

2.8 

8 

4.8 

4.0 

3-2 

9 

5-4 

4.5 

3.6 

0 

7 

6 

2 

5 

3 

4 

4 

3 

5 

2 

I 

0 

6 

37 

3.1 
9-2 
15-4 
21.6 
27.8 
33-9 


3.6 
10.8 
18.0 
25.2 
32-4 


5 
38 

3.8 
U.4 
19.0 
26.6 
34.2 


38 

4.8 
14.2 
23-8 
33-2 


37 

3-7 
II. I 

18.5 
25-9 
33-3 


37 

4.6 

139 
23.1 

32.4 


P.  P. 


68° 


48 


1 

L.  Sin. 

d. 

L.  Tan. 

c.  d. 

L.  Cot. 

L.  Cos. 

d. 



P.P. 

0 

I 

9-57358 

31 
31 
31 
31 
32 
31 
31 

9.60.641 

36 
36 

I 

36 
36 
36 

36 

3.6 
36 

^i 
36 

36 

^i 
36 
36 
36 
36 
36 
36 
35 
36 
36 
36 
35 
36 

^% 
36 

35 

36 

35 

36 

36 

35 

36 

35 
36 
35 
36 

35 
36 

35 
35 
36 
35 
35 
36 
35 

0-39  359 

9.96717 

6 

5 

60 

59 

9-57  389 

9.60  677 

0.39  323 

9.96711' 

2 

9.57420 

9.60714 

0.39  286 

9.96  706 

58 

37   36  35 

3 

9-57451 

9.60  750 

0.39  250 

9.96  701 

5 

5 
5 
5 
5 

5 
6 

5 

5 

57 

I 

3-7     3-6  3-5 

4 

9.57  482 

9.60  786 

0.39214 

9.96  696 

56 

2 

7-4  7-2  7.0 

S 

9-57  5H 

9.60  823 

0.39177 

9.96  691 

55 

3 

II. I  10.8  10.5 

6 

9-57  545 

9.60  859 

0.39  141 

9.96  686 

54 

4 

14.8  14.4  14.0 

7 

9-57576 

9.60  895 

0.39  105 

9.96  681 

S3 

5 

18.5  18.0  17.5 

8 

9.57  607 

31 

9.60931 

0.39  069 

9.96  676 

52 

6 

22.2  21.6  21.0 

9 
10 

II 

9-57  638 

31 
31 
31 
31 

9.60  967 

0.39  033 

9.96  670 

51 
50 

49 

7 
8 

9 

25.9  25.2  24.5 
29.6  28.8  28.0 
33-3   32.4  31-5 

9.57  669 

9.61  004 

0.38  996 

9.96  665 

9-57  700 

9.61  040 

0.38  960 

9.96  660 

12 

9-57  731 

9.61  076 

0.38  924 

9.96655 

5 

48 

13 

9-57  762 

31 
31 
31 
31 
30 
31 
31 
31 
30 

9.61  112 

0.38  888 

9.96  650 

5 

5 

5 
6 

5 
5 
5 
5 
6 

47 

H 

9-57  793 

9.61  148 

0.38  852 

9.96  645 

46 

32  31   30 

15 
16 

9-57  824 
9-57855 

9.61  184 
9.61  220 

0.38816 
0.38  780 

9.96  640 
9.96  634 

45 
44 

I 

2 

3.2  3-1  30 
6.4  6.2  6.0 

17 

9-57  §85 

9.61  256 

0.38  744 

9.96  629 

43 

3 

9.6  9-3  9.0 

18 

9.57916 

9.61  292 

0.38  708 

9.96  624 

42 

4 

12.8  12.4  12.0 

19 
20 

21 

9-57  947 

9.61  328 

0.38  672 

9.96  619 

41 
40 

39 

5 
6 

7 

16.0  15.5  15.0 
19.2  18.6  18.0 
22.4  21.7  21.0 

9-57978 

9.61  364 

0.38  636 

9.96  614 

9.58008 

9.61  400 

0.38  600 

9.96  608 

22 

9.58  039 

31 

9.61  436 

0.38  564 

9.96  603 

5 

38 

8 

25.6  24.8  24.0 

23 

9.58070 

31 
31 
30 
31 

9.61  472 

0.38  528 

9.96  598 

5 

37- 

9 

28.8  27.9  27.0 

24 

9.58  lOI 

9.61  508 

0.38  492 

9-96  593 

5 

36 

2S 

9-58  131 

9.61  544 

0.38  456 

9.96  588 

\ 
5 
5 
5 
5 
6 

5 
5 

35 

■ 

25 

9.58  162 

9.61  579 

0.38421 

9.96  582 

34 

29   6   5 

27 

9.58  192 

30 
31 

9.61  615 

0.38385 

9.96577 

33 

2.9  0.6  0.3 

5.8  1.2  I.O 

8.7  1.8  1.5 

1 1.6  2.4  2.0 

14.5  3.0  2.5 

28 

9.58223 

9.61  651 

0.38  349 

9.96572 

32 

2 

29 

30 

31 

9.58253 

.5° 
31 
30 
31 

9.61  687 

0.38313 

9.96  567 

31 
30 

29 

3 
4 

9.58  284 

9.61  722 

0.38  278 

9.96  562 

9.58  3'4 

9.61  758 

0.38  242 

9-96  556 

32 

958  345 

9.61  794 

0.38  206 

9-96551 

28 

6 

17.4  3.6  3.0 

Zl 

9-58375 

S^ 

9.61  830 

0.38  170 

9-96  546 

27 

7 

20.3  4-2  3-5 

34 

9.58  406 

31 

9.61  865 

0.38135 

9.96  541 

5 
6 

5 

5 

5 
6 

5 
5 
6 
5 
5 

\ 

26 

8 

23.2  4.8  4.0 

3S 

9-58  436 

30 
31 
30 
30 

9.61  901 

0.38  099 

9.96  535 

25 

9 

26.1  5.4  4.5 

36 

9.58467 

9.61  936 

0.38  064 

9.96  530 

24 

^^ 

9-58497 
9-58527 

9.61  972 

9.62  008 

0.38  028 
0.37  992 

9.96  525 
9.96  520 

23 

22 

39 
40 

41 

9.58  557 

30 
31 

30 

9.62  043 

0.37957 

9.96514 

21 
20 

19 

6    6 
36   35 

9-58  588 

9.62079 

0.37921 

9-96  509 

9.58618 

9.62  114' 

0.37  886 

9.96  504 

42 

9.58  648 

30 

9.62  150 

0.37  850 

9.96498 

18 

0 

3.0   2.9 

9.0   8.8 

15.0  14.6 

43 
44 

9-58678 
9.58  709 

30 

31 

9.62  185 
9.62  221 

0.37815 
0.37  779 

9-96  493 
9.96  488 

17 
16 

I 

2 

4S 

9-58  739 

30 
30 

9.62  256 

0.37  744 

9.96483 

15 

3 

21.0  20.4 

46 

9.58  769 

9.62  292 

0.37  708 

9.96477 

5 

14 

4 

27.0  26.2 

47 

9-58  799 

30 

9.62  327 

0.37  673 

9.96472 

13 

330  32.1 

48 

9.58829 

30 

9.62  362 

0.37  638 

9.96467 

12 

49 
50 

5'i 

9.58859 

30 
30 
30 

9.62  398 

0.37  602 

9.96  461 

5 
5 

II 
10 

9 

5   5   5 

9.58889 

9.62  433 

0.37  567 

9-96456 

9.58919 

9.62  468 

0.37  532 

9.96451 

S2 

9-58  949 

30 
30 

9.62  504 

0.37  496 

9-96  445 

5 

8 

37   36  35 

53 

9-58979 

9.62  539 

0.37461 

9.96  440 

7 

0 

54 

55 

9-59  009 
9-59  039 

3*^ 
30 

9.62574 
9.62  609 

3b 

% 

35 

35 
35 
35 

0.37  426 
0.37391 

9-96  435 
9.96429 

5 
6 

5 
5 
6 

5 
5 

6 

5 

I 
2 

3-7  3.6  3.5 
I  I.I  10.8  10.5 
18.5  18.0  17.5  1 
25.9  25.2  24.5 
33-3   32.4  31.5 

5^ 

57 
58 

9.59  069 

9-59  098 
9.59  128 

29 
30 

9.62  645 
9.62  680 
9.62715 

0.37  355 
0.37  320 
0.37  285 

9.96  424 
9.96419 
9.96413 

4 

3 

2 

3 
4 
5 

59 
60 

9-59  158 

30 
30 

9.62  750 

0.37  250 

9.96  408 

I 
0 

9-59  188 

9.62  785 

0.37215 

9-96  403 

L.  Cos. 

d. 

L.  Cot.  |c.  d. 

L.  Tan. 

L.  Sin. 

d. 

/ 

P.P. 

67° 


49 


L.  Sin. 


L.  Tan.  c.  d 


L.  Cot. 


L.  Cos. 


d. 


P.P. 


2 

3 

4 
5 
6 

7 
8 

9 

10 

II 

12 

13 
H 
15 
16 

17 
18 

19 

20 

21 

22 
23 

24 
25 
26 

27 
28 
29 
30 

31 

32 
33 
34 
35 
36 

37 
3B 
39 
40 

41 

42 

43 
44 
45 
46 

47 
48 

49 

50 

51 

52 
5.3 
54 
55 
56 

57 
58 
59 
60 


9.59  188 


9.59218 
9.59  247 
9-59277 
9-59  307 
9-59  33i^ 
9-59  366 
9-59  396 
9-59425 
9.59  451 


9-59  484 
9-59  5  H 
9-59  543 
9-59  573 
9.59  602 

9-59  ^32 
9.59651 

9.59  690 
9.59  720 
9.59  749 


9-59  778 


9.59  808 
9-59  837 
9-59  866 

9-59  895 
9-59  924 
9-59  954 

9-59983 
9.60012 
9.60041 


9.60070 


9.60  099 
9.60  128 
9.60  157 
9.60  186 
9.60  215 
9.60  244 
9.60  273 
9.60  302 
9.60331 


9.60359 


9.60  388 
9.60417 
9.60  446 
9.60  474 
9.60  503 
9.60  532 
9.60  561 
9.60  589 
9.60618 


9.60  646 


9.60  675 
9.60  704 
9.60  732 
9.60  761 
9.60  789 
9.60818 
9.60  846 
9.60  875 
9.60  903 


9.60931 


29 
30 
29 

30 
29 
30 
29 

30 
29 

29 
30 
29 

29 
30 

29 
29 

29 
29 
30 
29 
29 

29 
29 

29 
29 
29 
29 
29 
29 

29 
29 
29 
28 
29 

29 
29 

28 
29 
29 
29 

28 

29 
28 

29 

29 
28 

29 
28 
29 
28 

29 
28 

28 


9.62  785 


9.62  820 
9.62855 
9.62  890 
9.62  926 
9.62  961 

9.62  996 

9.63  031 
9.63  066 
9.63  lOI 


9»fa3  135 


9.63  170 
9.63  205 
9.63  240 
9.63  275 
9.63310 
9-63  34l 

9-63  379 
9.63414 

9-63  449 


9.63  484 


9.63519 

9.63  553 
9.63  588 

9.63  623 
9.63  657 
9.63  692 
9.63  726 
9.63  761 
9.63  796 


9.63  830 


9.63  865 
9.63  899 
9-63  934 

9.63  968 

9.64  003 
9.64  037 
9.64  072 
9.64  106 
9.64  140 


9-64175 


9.64  209 
9.64  243 
9.64  278 
9.64312 
9.64  346 
9.64  381 
9.64415 
9.64  449 
9.64  483 


9.64517 


9.64552 
9.64  586 
9.64  620 
9.64  654 
9.64  688 
9.64  722 
9.64  756 
9.64  790 
9.64  824 


9.64858 


35 
35 

35 

36 

35 

35 

35 

35 

35 

34 

35 

35 

35 

35 

35 

35 

34 

35 

35 

35 

35 

34 

35 

35 

34 

35 

34 

35 

35 

34 

35 

34 

35 

34 

35 

34 

35 

34 

34 

35 

34 

34 

35 

34 

34 

35 

34 

34 

34 

34 

35 

34 

34 

34 

34 

34 

34 

34 

34 

34 


0.37215 


9.96  403 


0.37  180 

0.37  M5 
0.37  no 

0.37  074 

0.37  039 
0.37  004 

0.36  969 
0.36  934 
0.36  899 


9.96  397 
9.96  3^2 
9.96  387 
9.96  381 
9.96  376 
9.96  370 

9-96  365 
9.96  360 

9-96  354 


o.36't>65 


9.96  349 


0.36  830 

0.36  795 
0.36  760 

0.36  725 
0.36  690 
0.36655 
0.36  621 
0.36  586 
0-36551 


9-96  343 
9.96  33^ 
9-96  333 
9.96327 
9.96  322 
9.96316 
9.96  311 
9.96  305 
9.96  300 


036510 


0.36481 
0.36  447 
0.36412 

0.36  377 
0.36  343 
0.36  308 

0.36  274 
0.36  239 
0.36  204 


0.36  170 


0.36135 
0.36  lOI 
0.36  066 
0.36032 
0-35  997 
0.35  963 
0.35  928 

0.35  894 
0.35  860 


0.35  825 


0.35  791 
0.35  757 
0.35  722 

0.35  688 

0.35  654 
0.35  619 

0.35  585 
0-35551 
0-35517 
0-35  483 


0.35  448 
0.35414 
0.35  380 
0.35  346 
0.35312 
0.35  278 
0.35  244 
0.35  210 
0.35  176 


0.35  H2 


9-96  294 
9.96  289 
9.96  284 
9.96  278 
9.96  273 
9.96  267 
9.96  262 
9.96  256 
9.96  251 
9.96  245 


9.96  240 


9.96  234 
9.96  229 
9.96  223 
9.96  2l8 

9.96  212 
9.96  207 
9.96  201 
9.96  196 
9.96  190 


9-96  185 


9.96179 
9.96174 
9.96  168 
9.96  162 
9.96157 
9-96151 
9.96  146 
9.96  140 
9-96  135 
9.96  129 


9.96  123 
9.96  118 
9.96  112 
9.96  107 
9.96  lOI 
9.96  095 
9.96  090 
9.96  084 
9^96  079 
9-96  073 


60 

59 
58 
57 
56 
55 
54 

53 
52 
51 
50 

49 
48 

47 
46 
45 
44 

43 
42 
41 
40 

39 
3S 
37 
36 
35 
34 

33 
32 
31 
30 

29 
28 

27 
26 

25 
24 

23 
22 
21 

20 

19 
18 

17 
16 

15 
14 

13 
12 
II 

10 

9 
8 

7 
6 
5 
4 

3 
2 
I 
O 


35      34 


2 

3.6     3.5 
7.2     7.0 

3-4 
6.8 

3 

10.8  10.5 

10.2 

4 
5 
6 

144  M-o 
18.0  17.5 
21.6  21.0 

13.6 
17.0 
204 

7 
8 

25.2  24.5 
28.8  28.0 

23.8 
27.2 

9 

324  3' -5 

30.6 

I 
2 

3-0 
6.0- 

2.9 
S.8 

3 
4 

9.0 
12.0 

.  8.7 
11.6 

5 
6 

7 

15.0 
18.0 
21.0 

14-5 
17.4 
20.3 

8 
9 

24.0 
27.0 

23.2 
26.1 

6 

0.6 
1.2 
1.8 
24 
3-0 
3-6 
4.2 
4.8 
5.4 


29     28 

2.8 
5.6 
84 
II. 2 
14.0 
16.8 
19.6 
22.4 
25.2 


0.5 
i.o 

1-5 
2.0 

2.5 
3-0 

3.5 
4.0 

4-5 


6 
36 


35 


2.9 
8.8 


3.0 

9.0  5.5     5.5 

15.0  14.6  14.2 

21.0  204  19.8 

27.0  26.2  25.5 

33.0  32.1  31.2 


35 

3.5 
10.5 

17-5 
24-5 
31-5 


34 

34 
10.2 
17.0 
23.8 
30.6 


I   L.  Cos.    i   d.  I    L.  Cot,     c.  d.    L.  Tan. 


L.  Sin. 


d.  I 


P.  P. 


50 


L.  Sin.   d. 


L.  Tan.  |c.  d. 


L.  Cot. 


L.  Cos. 


d. 


P.P. 


O 

I 

2 

3 
4 
5 
6 

7 
8 

9 

10 

II 

12 

13 
H 
^5 
i6 

17 
18 

19 

20 

21 

22 
^3 

24 

25 
26 

27 
28 
29 

30 

31 

32 
33 
34 
35 
36 

37 
38 
39 
40 

41 

42 
43 
44 
45 
46 

47 
48 

49 

50 

51 
52 
53 
54 
55 
56 

57 
58 
59 
60 


9.60931 


9.60  960 

9.60  988 

9.61  016 
9.61  045 
9.61  073 
9,61  lOI 
9.61  129 
9.61  158 
9.61  186 


9.61  214 


9.61  242 
9.61  270 
9.61  298 
9.61  326 
9.61  354 
9.61  382 
9.61  411 
9.61  438 
9.61  466 


9.61  494 


9.61  522 
9.61  550 
9.61  578 
9.61  606 
9.61  634 
9,61  662 
9.61  689 
9.61  717 
9.61  745 


9.61  773 


9.61  800 
9.61  828 
9.61  856 
9.61  883 
9.61  911 
9.61  939 
9.61  966 
9.61  994 
9.62021 


9.62  049 


9.62076 
9.62  104 
9.62  131 
9.62  159 
9.62  186 
9.62  214 
9.62  241 
9.62  268 
9.62  296 


9.62  323 


9.62  350 
9.62  377 
9.62  405 
9.62  432 
9.62459 
9.62  486 
9.62513 
9.62  541 
9.62  568 


9-62  595 
L.  Cos. 


9.64 


9.64  892 
9.64  926 
9.64  960 

9.64  994 

9.65  028 
9.65  062 
9.65  096 
9.65  130 
9.65  164 


9.65  197 


9.65  231 
9.65  265 
9.65  299 

9.65  333 
9.65  366 
9.65  400 

9-65  434 
9.65  467 
9.65501 


9-65  535 


9.65  568 
9.65  602 
9.65  636 
9.65  669 
9.65  703 
9.65  736 
9.65  770 
9.65  803 
9-65  837 


9.65  870 


9.65  904 

9-65  937 

9.65  971 

9.66  004 
9.66038 
9.66071 
9.66  104 
9.66138 
9.66  171 


(.66  204 


9.66  238 
9.66  271 
9.66  304 

9-66  337 
9.66371 
9.66  404 
9.66437 
9.66  470 
9-66  503 


9-66  537 


9.66  570 
9.66  603 
9.66  636 
9.66  669 
9.66  702 
9-66  735 
9.66  768 
9.66  801 
9.66  834 


9.66  867 
L.  Cot. 


34 
34 
34 
34 
34 
34 
34 
34 
34 
33 
34 
34 
34 
34 
33 
34 
34 
33 
34 
34 
33 
34 
34 
33 
34 
33 
34 
33 
34 
33 
34 
33 
34 
33 
34 
33 
33 
34 
33 
33 
34 
33 
33 
33 
34 
33 
33 
33 
33 
34 
33 
33 
33 
33 
33 
33 
33 
33 
33 
33 

cTd 


0.35  142 


9.96073 


0.35  108 
0.35  074 
0.35  040 
0.35  006 
0.34972 
0.34  938 
0.34  904 
0.34  870 
0.34  836 


9.96067 
9.96062 
9.96056 
9.96050 
9.96  045 
9.96039 
9.96  034 
-9.96  028 
9.96  022 


0.34  803 


9.96017 


0.34  769 
0-34  735 
0.34  70 i 
0.34  667 

0.34  634 
0.34  600 

0.34  566 

0.34  533 
0.34  499 


9.96  01 1 
9.96  005 
9.96  000 

9.95  994 
9.95  988 
9.95  982 

9.95  977 
9-95971 
9-95  965 


0.34  465 


9.95  960 


0.34  432 
0.34  398 
0.34  364 

0.34331 
0.34  297 
0.34  264 
0.34  230 

0.34  197 
0.34  163 


9-95  954 
9.95  948 

9-95  942 
9-95  937 
9-95  931 
9-95  925 
9.95  920 

9-95  9H 
9.95  908 


0.34  130 


9.95  902 


0.34  096 
0.34  063 
0.34  029 
0.33  996 
0.33  962 
0.33  929 
0.33  896 
0.33  862 
0.33  829 


9-95  897 
9.95  891 

9.95  885 

9.95  879 

•9-95  873 

9.95  868 

9.95  862 
9-95  856 
9-95  850 


0.33  796 


9-95  844 


0.33  762 

0.33  729 
0.33  696 

0.33  663 
0.33  629 
0.33  596 
0.33  563 
0.33  530 
0.33  497 


9-95  839 
9-95  833 
9.95  827 

9.95821 

9-95  815 
9.95  810 

9.95  804 
9-95  798 
9-95  792 


0.33  46: 


•95  786 


0.33  430 
0.33  397 
0.33  364 

0-33  33^ 
0.33  298 
0.33  265 
0.33  232 

0.33  199 
0.33  166 


9.95  780 
9-95  775 
9-95  769 
9-95  763 
9-95  757 
9-95751 
9-95  745 
9-95  739 
9-95  733 


0-33  ^33 
L.  Tan. 


9-95  728 
L.  Sin. 


60 

59 
58 
57 
56 
55 
54 

53 
52 
51 
50 

49 
48 
47 
46 

45 
44 

43 
42 

41 

40 

39 
38 
37 
36 
35 
34 

33 
32 
31 
30 

29 
28 
27 
26 
25 
24 

23 
22 
21 
20 

19 
18 

17 
16 
15 
14 

13 
12 
II 

10 


34 

3-4 
6.8 
10.2 
13-6 
17.0 
20.4 
23-8 
27.2 
30.6 


33 

3-3 
6.6 
9.9 
13.2 
16.5 
19.8 
23.1 
26.4 
29.7 


29  28  27 


I 

2.9 

2.8 

2.7 

2 

5.8 

5.6 

S-4 

3 

8.7 

8.4 

8.1 

4 

11.6 

II. 2 

10.8 

s 

14.5 

14.0 

1.^.5 

6 

17.4 

16.8 

16.2 

7 

20.3 

19.6 

18.9 

8 

23.2 

22.4 

21.6 

9 

26.1 

25.2 

24.3 

6 

0.6 
1.2 
1.8 

2.4 

30 

3-6 
4.2 

4.8 
5-4 


0-5 
i.o 

1-5 
2.0 

2-5 

30 

3-5 
4.0 

4-5 


6^ 
34 

2.8 
8.5 


33 


2.5 
8.2 
14.2  13.8 
19.8  19.2 


_5^ 
34 

3-4 
10.2 
17.0 

2^.s 


25.5  24.8  30.6 
31.2  30.2  — 


p.p. 


a^° 


5' 


O 

I 

2 

3 
4 
5 
6 

7 
8 

9 

10 

II 

12 

13 
H 
15 
16 

17 
18 

19 

20 

21 

22 
23 
24 

25 
26 

27 
28 
29 

30 

31 
32 
33 
34 
35 
36 

31 
38 
39 
40 

41 
42 
43 
44 
45 
46 

47 
48 

49 

50 

51 

52 
53 
54 
55 
56 

57 
58 
59 
60 


L.  Sin. 
9.62  595 


9.62  622 
9.62  649 
9.62  676 
9.62  703 
9.62  730 
9.62  757 
9.62  784 
9.62  81 1 
9.62  838 


9.62  86g 
9.62  892 
9.62918 
9.62  945 
9.62972 

9.62  999 

9.63  026 
9.63052 
9.63  079 
9.63  106 


9-63  133 


9.63  159 
9.63  186 
9.63213 
9.63  239 
9.63  266 
9.63  292 

963319 
9-63  345 
9-63  372 


9-63  398 


9.63  425 
9.63451 
9.63  478 

9-63  504 
9-63  531 
9-63557 

963  583 
9.63  610 
9.63  636 


9.63  662 


9.63  689 
9-63715 
9-63  741 
9.63  767 
9-63  794 
9.63  820 

9.63  846 
9.63  872 
9.63  898 


•63  924 


9-63  950 

9.63  976 

9.64  002 
9.64028 
9.64054 
9.64  080 
9.64  106 
9.64  132 
9.64  158 


9.64  184 


L.  Tan. 

9.66  867 
9.66900 
9-66  933 
9.66  966 

9.66  999 

9.67  032 
9.67  065 
9.67  098 

9.67  131 
9.67  163 


9.67  196 


9.67  229 
9.67  262 
9.67  295 
9.67  327 
9-67  360 
9-67  393 
9.67  426 
9.67  458 
9.67  491 


9.67  524 


9.67  556 
9.67  589 
9.67  622 
9.67  654 
9.67  687 
9.67  719 
9.67  752 

9.67  785 
9.67  817 


9.67  850 


9.67  882 
9.67915 
9-67  947 

9.67  980 
9,68012 

9.68  044 
9.68077 
9,68  109 
9.68  142 


9.68  174 


9.68  206 
9-68  239 
9.68  271 
9.68  303 
9.68336 
9.68  368 
9.68  400 
9.68  432 
9.68  465 


9.68  497 


9.68  529 
9.68  561 
9-68  593 
9.68  626 
9.68  658 
9.68  690 
9.68  722 
9.68  754 
9.68  786 


9.68818 


c.  d 

32, 
33 
33 
33 
33 
33 
33 
33 
32 
33 
33 
33 
33 
32 
33 
33 
33 
32 
33 
33 
32 
33 
33 
32 
33 
32 
33 
33 
32 
33 
32 

33 
32 
33 
32 
32 
33 
32 
33 
32 
32 
33 
32 
32 
33 
32 
32 
32 
33 
32 
32 
32 
32 
33 
32 
32 
32 
32 
32 
32 


L.  Cot. 

^13J33^ 
0.33  too 
0.33067 
0.33  034 
0.33001 
0.32  968 
0.32  935 
0.32  902 
0.32  869 
0.32837 


0.32  804 


0.32771 
0.32  738 
0.32  705 

0.32  673 
0.32  640 
0.32  607 

0.32  574 
0.32  542 
0-32  509 


0.32  476 


0.32  444 
0.32  411 
0.32  378 
0.32  346 

0.32313 
0.32  281 

0.32  248 
0.32  215 
0.32  183 


0.32  150 


0.32  118 
0.32085 
0.32053 
0.32  020 
0^1  988 
0.31  956 
0.31  923 
0.31  891 
0.31  858 


0.31  826 


0.31  794 
0.31  761 
0.31  729 
0.31  697 
0.31  664 
0.31  632 
0.31  600 
0.31  568 
0-31  535 


0.31  503 


0.31  471 

0.31  439 
0.31  407 

0.31  374 
0.31  342 
0.31  310 
0.31  278 
0.31  246 
0.31  214 


0.31  182 


L.  Cos. 

995  728 


9-95  722 
9-95  716 
9-95  710 
9.95  704 
9.95  698 
9.95  692 

9.95  686 
9.95  680 
9-95  674 


9-95  668 


9.95  663 
9.95  657 
9-95  651 
9-95  64I 
9-95  639 
9-95  633 
9-95  627 
9.95  621 

995615 


9.95  609 


9.95  603 
9-95  597 
9-95  591 
9.95  585 
9-95  579 
9-95  573 
9-95  567 
9-95  561 
9-95  555 


9-95  549 


9-95  543 
9-95  537 
9-95  531 
9-95  525 
9-95519 
9-95513 
9-95  507 
9-95  500 
9-95  494 


9.95  488 


9.95  482 
9-95  476 
9-95  470 
9-95  464 
9-95  458 
9-95  452 

9-95  446 
9.95  440 

9-95  434 


9-95  427 


9-95  421 
9-95415 
9.95  409 

9-95  403 
9-95  397 
9-95  391 
9.95  384 
9-95  378 
9-95  372 


9-95  366 


60 

59 
58 
57 
56 
55 
54 

53 
52 
51 
50 

49 
48 
47 
46 
45 
44 

43 
42 
41 
40 

39 
3^ 
37 
36 
35 
34 

33 
32 
31 
30 

29 
28 
27 
26 
25 
24 

23 
22 
21 

20 

19 

18 

17 
16 
15 
14 

13 
12 
II 

10 

9 
8 

7 
6 
5 
4 

3 
2 
I 
O 


P.  P. 


33   32 


3-2 

6.4 
9.6 
12.8 
16.0 
19.2 
22.4 
25.6 
28.8 


27   26 


I 

.3.3 

2 

6.6 

3 

9-9 

4 

13.2 

5 

16.5 

6 

19.8 

7 

23.1 

8 

26.4 

9 

29.7 

2-7 

5-4 

8.1 

10.8 

13-5 
16.2 
18.9 
21.6 
243 


2.6 
5-2 
7-8 
10.4 
13.0 
15.6 
18.2 
20.8 
23-4 


I 

0.7 

0.6 

0.5 

2 

1.4 

1.2 

I.O 

3 

2.1 

1.8 

1.5 

4 

2.8 

2.4 

2.0 

5 

3.5 

3-0 

2-5 

6 

4.2 

3.6 

3-0 

7 

4-9 

4.2 

3-5 

8 

5.6 

4-8 

4.0 

9 

6.3 

5-4 

4-5 

32 

2-3 
6.9 
11.4 
16.0 
20.^ 
25.1 
29.7 


32 

2.7 
8.0 

13-3 

18.7 
24.0 
29-3 


33 

3-3 

9-9 

16.5 

23.1 

29.7 


L.  Cos. 


L.  Cot. 


c.  d. 


L.  Tan- 
fid." 


L.  Sin.  I  d. 


P.P. 


/ 

L.  Sin. 

d. 

L.Tan.  c.  d. 

L.  Cot.  1  L.  Cos. 

d.| 

P.  P.              1 

0 

I 

9.64  184 

26 
26 
26 

9.68818 

32 

0.31  182 

9.95  366 

6 
6 

60 

59 

9.64  210 

9.68  850 

0.31  150 

9.95  360 

2 

9.64  236 

9.68  882 

32 
32 

0.31  118 

9-95  354 

58 

3 

9.64  2b2 

26 

25 
26 

9.68914 

0.31  086 

9-95  348 

57 

4 

9.64  288 

9.68946 

32. 

32 
10 

0.31  054 

9-95  341 

7 
5 

56 

32    31 

5 

964313 

9.68978 

0.31022 

9-95  335 

6 

55 

I 

3-2   3.1   1 

6 

9-64  339 

26 
26 
26 
25 
26 
26 

9.69  OIO 

32 
32 
32 
32 
32 
32 
32 
32 
32 
31 

0.30  990 

9-95  329 

6 

6 

7 
6 

6 
6 

54 

2 

6.4   6.2 

7 

9.64  365 

9.69  042 

0.30  958 

9-95  323 

53 

3 

9-6   9-3 

8 

9.64391 

9.69  074 

0.30  926 

9-95317 

52 

4 

12.8   12.4 

9 
10 

II 

9.64  4P7 

9.69  106 

0.30  894 

9-95  310 

51 
50 

49 

'  5 
6 

7 
8 

16.0   15.5 
19.2   18.6 
22.4   21.7 
25.6   24.8 

9.64  442 

9.69  1.38 

0.30  862 

9-95  304 

9.64  468 

9.69  170 

0.30  830 

9.95  298 

12 

9.64  494 

9.69  202 

0.30  798 

9.95  292 

A 

48 

9 

28.8   2-7.0 

13 

9.64519 

25 
26 
26 

9-69  234 

0.30  766 

9.95  286 

47 

14 

9-64  545 

9.69  266 

0.30  734 

9.95  279 

7 

46 

IS 

9-64571 

9.69  298 

0.30  702 

9-95  273 

6 
6 

45 

i6 

9.64  596 

25 
26 

11 

25 
26 

9.69  329 

0.30671 

9.95  267 

44 

17 

9.64622 

9.69  361 

32 

0.30  639 

9-95  261 

43 

26  25  24 

18 

9.64  647 

9-69  393  C 

0.30  607 

9-95  254 

7 

t 
6 

42 

I 

2.6  2.5  2.4  1 

19 
20 

21 

9.64  673 

9.69  425 

32 
31 

0-30  575 

9-95  248 

41 
40 

39 

2 
3 

4 

5.2  5.0  4.8 

7-8  7-5  7-2 
10.4  lO.O  9.6 

9.64  698 

9-69  457 

'^.30  543 

9.95  242 

9.64  724 

9.69  488 

0.30  5 1 2 

9.95  236 

22 

9.64  749 

25 
26 

25 
26 

9.69  520 

32 
32 
32 
31 
32 
32 
31 
32 
32 
31 
32 
31 
32 
32 
31 
32 
31 
•32 
31 
32 

31 
32 

0.30  480 

9.95  229 

7 
6 

6 
6 

3^ 

7 
8 

13.0  12.5  12.0  1 
15.6  15.0  14.4 
18.2  17.5  16.8 

20.8  20.0  19.2  ] 

23 
24 

9-64  775 
9.64  800 

9.69  552 
9.69  584 

0.30  448 
0.30416 

9.95  223 
9-95217 

37 
36 

2.S 

9.64826 

9.69615 

0.30385 

9.95  211 

7 
6 

35 

g 

22.A    22. C  21.6  1 

25 

9.64851 

26 

9.69  647 

0.30  353 

9-95  204 

34 

27 

9.64877 

9.69  679 

0.30321 

9.95  198 

33 

28 

9.64  902 

25 

9.69710 

0.30  290 

9.95  192 

32 

29 

9.64927 

^5 

9.69  742 

0.30  258 

9.95  185 

7 
6 

31 

30 

31 

9.64  953 

25 

9.69  774 

0.30  226 

9-95  179 

I 

7 

A 

30 

29 

7    6 

9.64  978 

9.69  805 

0.30  195 

9-95  173 

1 

0.7    0.6 

32 

9.65  003 

'I 

9.69  837 

0.30  163 

9.95  167 

28 

>   1.4    1.2 

33 

9.65  029 

9.69  868 

0.30132 

9-95  160 

27 

1  1  2.1    1.8 

34 

9-65  054 

25 

9.69  900 

0.30.100 

9-95  154 

6 

26 

^  2.8  2.4 

35 

9.65  079 

25 

9.69932 

0.30  068 

9-95  148 

7 
6 
6 

25 

)  3-5   3-0 

36 

9.65  104 

25 
26 

25 

9.69  963 

0.30037 

9-95  141 

24 

)  4.2   3.6 

37 

9.65  130 

9-69  995 

0.30  005 

9-95  135 

23 

f    4-9   4-2 
I    5.6   4.8 
)  6.3   5.4 

18 

9-65  155 

9.70026 

0.29  974 

9-95  129 

22 

39 
40 

41 

9.65  180 

25 
25 
25 

25 
26 

9.70058 

0.29  942 

9.95  122 

7 
6 

6 

7 
6 

21 
20 

19 

9.65  205 

9.70  089 

0.29  911 

9.95  116 

9.65  230 

9.70121 

0.29  879 

9-95  "o 

42 

9-65  255 
9.65  281 

9.65  306 

9.70152 
9.70  184 
9.70215 

0.29  848 
0.29  816 
0.29  785 

9.95  103 
9-95  097 
9.95  090 

18 
17 
16 

43 
44 

25 
25 
25 

31 
32 
31 
31 
32 
31 
32 

31 
31 

32 

7 
6 

4S 

9-65  33i 

9.70  247 

0.29  753 

9-95  084 

6 

15 

46 

9-65  356 

9.70278 

0.29  722 

9-95  078 

7 
6 

H 

47 

9.65  381 

25 
25 

i 

9.70  309 

0.29  691 

9.95071 

13 

7   7   6 

48 
49 
50 

51 

9.65  406 
9-65431 

9.70341 
9.70372 

0.29  659 
0.29  628 

9.95  065 
9-95  059 

6 
7 
6 

7 

12 
II 

10 

9 

32  31   32 

9.65450 

9.70  404 

0.29  596 

9-95052 

I 
2 

2.3  2.2  2.7 
6.9  6.6  8.0 

9.65  481 

9-70  435 

0.29  565 

9-95  046 

52 

9.65  506 

25 

9.70466 

0.29  534 

9-95  039 

8 

11.4  II. I  13.3 

53 

9-65  531 

25 

9.70498 

0.29  502 

9-95  033 

6 

7 
6 

7 

4 

S 

16.0  15.5  18.7 

54 

9.65  556 

23 

24 
25 
25 
25 
25 
25 

9-70529    i[ 

0.29471 

9.95  027 

6 

20.6  19.9  24.0 
25.1  24.4  29.3 

29.7  28.8  — 

55 

9.65  580 

9-70560  1  ^2 
9.70592  i  ^^ 

0.29  440 

9.95  020 

5 

6 

56 

9.65  605 

0.29  408 

9.95014 

; 

4 

7 

57 

9.65  630 

9.70  623 

31 
31 

0.29  377 

9-95  007 

3 

58 

9-65  655 

9.70654 

0.29  346 

9.95  001 

6 

2 

59 
60 

9.65  680 

9.70  685 

0.29315 

9-94  995 

' 

I 
0 

9-65  705 

9.70717 

0.29  283 

9.94  988 

1  L.  Cos. 

1  d. 

L.  Cot.  |c.  d.|  L.Tan.  |  L.  Sin. 

1  d.  1  '  1     P.  P.     1 

63' 


53 


/ 

L.  Sin.   d.  1 

L.Tan. 

c.  d.| 

L.  Cot.   L.  Cos.  1 

d. 

p.p.       il 

o 

I 

9.65  705 

24 
25 

25 

9.70717 

31 
31 
31 

0.29  283 

9.94  988 

6 

7 
6 

60 

59 

9.65  729 

9.70  748 

0.29  252 

9.94  982 

2 

9-65  754 

9.70  779 

0.29  221 

9-94  975 

58 

3 

965  779 

9.70810 

0.29  190 

9.94  909 

57 

4 

9.65  804 

-^5 
24 

9.70  841 

31 

32 
31 
31 

0.29  159 

9.94  962 

7 
6 

56 

32  31  30 

5 

9.65  828 

9.70873 

0.29  127 

9.94  956 

7 
6 

55 

I 

32  3.1  30 

6 

9-65  853 

25 

25 

9.70  904 

0.29  096 

9.94  949 

54 

2 

6.4  6.2  6.0 

7 

9.65  878 

970935 

0,29  065 

9-94  943 

7 
6 

53 

3 

9.6  9.3  9.0 
12.8  12.4  12.0 
16.0  15.5  15.0 
19.2  18.6  18.0 
22.4  21.7  21.0 
25.6  24.8  24.0 
28.8  27.9  27.0 

8 

9.65  902 

24 
25 
25 
24 
25 
24 

9.70966 

31 
31 
31 
31 
31 
31 

0.29  034 

9-94  936 

52 

4 

9 
10 

12 

13 

9.65  927 

9.70997 

0.29  003 

9.94  930 

7 
6 
6 

7 
6 

7 
6 

51 
50 

49 
48 
47 

7 
8 

9 

9-65  952 

9.71  028 

0.28  972 

9-94923 

9.65  976 
9.66001 
9.66025 

9.71059 
9.71090 

9.71  121 

0.28  941 
0.28910 
0.28  879 

9-94917 
9.94  91 1  1 
9.94  904 

H 

9.66  050 

25 
25 
24 

9-71  153 

32 
31 
31 

0.28  847 

9-94  898 

46 

IS 

9.66075 

9.71  184 

0.28816 

9.94891 

45 

i6 

9.66  099 

9.71  215 

0.28  785 

9-94  88-5 

44 

17 

9.66  124 

24 

9.71  246 

31 
31 
31 
31 
31 

0.28  754 

9.94878  1 

7 

43 

25   24  23 

18 

9.66  148 

9.71  277 

0.28  723 

9-94871  '  k   1 

42 

I 

2.5  2.4  2.3 

19 
20 

21 

9.66173 

25 
24 
24 

9.71  308 

0.28  692 

9.94865 

7 
6 

41 
40 

39 

2 
3 
4 

5.0  4.8  4.6 

7.5  7.2  6.9 

lo.o  9.6  9.2 

9,66  197 

9.71  339 

0.28  661 

9.94  858 

9.66  221 

9.71  370 

0.28  630 

9-94  852 

22 

9.66  246 

^5 

9.71  401 

31 

0.28  599 

9-94  845 

7 
6 

7 
6 

38 

I 

12.5  12.0  11.5 
15.0  14.4  13.8 
17.5  16.8  16.1 
20.0  19.2  18.4 

23 

24 

9.66  270 
9-66  295 

24 
25 
24 
24 

25 

9-71  431 
9.71  462 

30 
31 
31 
31 
31 
31 
31 
31 
31 
30 
31 

0.28  569 
0.28  538 

9-94  839 
9-94  832 

37 
36 

25 

9.66319 

971  493 

0.28  507 

9.94  826 

7 
6 

35 

9 

22. c  21.6  20.7 

26 

9-66  343 

971  524 

0.28476 

9.94819 

34 

27 

9.66  368 

9.71  555 

0.28  445 

9.94813 

7 
7 
6 

7 
6 

7 
6 

7 

I 

7 
5 

33 

28 

9.66  392 

24 

9.71  586 

0.28414 

9.94  806 

32 

29 
30 

31 

9.66416 

24 
25 
24 

9.71617 

0.28  383 

9.94  799 

31 
30 

29 

7   6 

9.66441 

9.71  648 

0.28  352 

9-94  793 

9.66  465 

9.71  679 

0.28321 

9-94  786 

I 

0.7  0.6 

32 

9.66  489 

24 
24 

9.71  709 

0.28  291 

9.94  780 

28 

2 

1.4  1.2 

33 

9.66513 

9.71  740 

0.28  260 

9-94  773 

27 

3 

2.1  1.8 

34 

9.66  537 

24 

9.71  771 

31 
31 

0.28  229 

9.94  767 

26 

4 

2.8  2.4 

35 

9.66  562 

25 

9.71  802 

0.28  198 

9.94  760 

25 

^. 

'■^     ^-A 

36 

9.66  586 

24 

9-71  833 

31 

0.28  167 

9-94  753 

24 

6 

4.2  3.6 

37 

9.66610 

24 

9.71  863 

3^ 
31 
31 
30 
31 
31 
31 
30 
31 
31 

0.28  137 

9-94  747 

23 

I 

9 

4.9  4.2 
5.6  4-8 
6.3  5-4 

38 

9.66  634 

24 
24 
24 

24 

25 
24 
24 

24 

9.71  894 

0.28  106 

9-94  740, 

22 

39 
40 

41 

9.66  658 

9.71925 

0.28075 

9-94  734 

7 

7 
6 

21 
20 

19 

9.66  682 

971955 

0.28  045 

9.94  727 

9.66  706 

9.71  986 

0.28014 

9.94  720 

42 
43 

9.66731 
9.66  755 

9.72017 
9.72048 

0.27  983 
0.27952 

9.94714 
9.94  707 

7 

7 
6 

18 
17 

44 

9.66  779 

9.72078 

0.27  922 

9.94  700 

16 

45 

9.66  803 

9.72  109 

0.27  891 

9.94  694 

7 

7 
(3 

15 

46 

9.66  827 

24 

9.72  140 

0.27  860 

9.94  687 

14 

47 

9.66851 

24 
24 
24 
23 
24 

9.72  170 

30 
31 
30 
31 
31 
30 
31 
30 
31 
30 
31 
30 
31 
30 

0.27  830 

9.94  680 

13 

7   6   6 

48 

9.66  875 

9.72  201 

0.27  799 

9.94  674 

7 
7 
6 

7 
7 
6 

7 
7 
6 

12 

30   31   30 

49 
50 

SI 

9.66  899 

9.72231 

0.27  769 

9-94  667 

II 

10 

9 

0 
2 

2.1   2.6   2.5 

6.4  7-8  7.5 
10.7  12.9  12.5 
15.0  18.1  17.5 
19.3  23.2  22.5 
23.6  28.4  27.5 

9.66  922 

9.72  262 

0.27  738 

9.94  660 

9.66  946 

9.72  293 

0.27  707 

9-94  654 

52 

9.66  970 

24 

9-72323 

0.27  677 

9.94  647 

8 

3 

53 
S4 

9.66  994 
9.67018 

24 
24 
24 
24 
24 
23 

972354 
972384 

0.27  646 
0.27616 

9.94  640 
9-94  634 

7 
6 

4 
5 
6 

ss 

9.67  042 

972415 

0.27585 

9.94  627 

5 

27.9  —  — 

5^ 

9.67066 

972445 

0-27555 

9.94  620 

4 

7 

57 

9.67  090 

9.72476 

0.27  524 

9.94614 

7 
7 
7 

3 

SB 

9.67  113 

9.72  506 

0.27  494 

9.94  607 

2 

59 
60 

9.67  137 

24 
-  24 

972  537 

0.27  463 

9.94  600 

I 
0 

9.67  161 

972  567 

0.27  433 

9.94  593 

L.  Cos. 

d. 

L.  Cot.  |c.  d 

.  L.Tan. 

1  L.  Sin.  1  d. 

1  '        P.P. 

£>Oo 


54 


/ 

L.  Sin. 

d. 

L.  Tan. 

c.  d.|  L.  Cot. 

L.  Cos. 

d. 

P.P. 

0 

I 

9.67  161 

24 
23 

9.72567 

31 
30 
31 

0.27  433 

9-94  593 

6 

60 

59 

9.67  185 

9.72  598 

0.27  402 

9-94  587 

2 

9.67  208 

9.72  628 

0.27  372 

9.94  580 

58 

3 

9.67  232 

24 

9-72659 

0.27  341 

9-94  573 

7 
6 

57 

4 

9.67  256 

24 

9.72  689 

30 
31 
30 
30 

31 
30 
31 
30 
30 
31 
30 
30 
31 
30 
30 
30 
31 
30 
30 

0.27  311 

9-94  567 

56 

31   30  29 

I 

9.67  280 
9-67  303 

24 
23 
24 

9.72  720 
9.72  750 

0.27  280 
0.27  250 

9.94  560 
9-94  553 

55 
54 

2 

3.1  3-0  2.9 

6.2  6.0  5.8 
9-3  9-0  8.7 

7 

9.67  327 

9.72  780 

0.27  220 

9-94  546 

53 

3 

8 

9.67  350 

^0 
24 
24 
23 

9.72  811 

0.27  189 

9-94  540 

52 

4 

12.4  12.0  1 1.6 

9 
10 

II 

967  374 

9.72841 

0.27  159 

9-94  533 

51 
50 

49 

5 
6 

7 
8 

15-5  15-0  14-5 

18.6  18.0  17.4 

21.7  21.0  20.3 

9.67  398 

9.72872 

0.27  128 

9-94  526 

9.67421 

9.72902 

0.27  098 

9-94519 

12 

9-67445 

24 

9.72932 

0.27  068 

9-94513 

48 

24.8  24.0  23.2 

13 

9.67  468 

23 
24 
23 
24 

9.72963 

0.27037 

9-94  506 

47 

9 

27.9  27.0  26.1 

H 

9.67  492 

9-72993 

0.27007 

9-94  499 

46 

15 

9-67515 

9-73023 

0.26977 

9-94  492 

45 

lb 

9-67  539 

9-73054 

0.26  946 

9-94485 

44 

17 

9.67  562 

23 

24 

9-73084 

0.26916 

9-94  479 

43 

24  23  22 

18 

9.67  586 

9-73114 

0.26  886 

9-94  472 

42 

19 
20 

21 

9.67  609 

23 

It 

9-73  144 
9-73  175 

0.26856 

9-94465 

41 
40 

39 

2 
3 

2.4  2.3  2.2 
4.8  4.6  4.4 

7.2  6.9  6.6 
9.6  9-2  8.8 

9-67  633 

0.26  825 

9-94  458 

9.67  656 

9-73  205 

0.26  795 

9-94451 

22 

9.67  680 

24 

9-73  235 

0.26  765 

9-94  445 

S8 

4 

23 

24 

9.67  703 
9.67  726 

^3 
23 

9-73  265 
9-73  295 

30 
30 
31 
30 
30 
30 

0.26  735 
0.26  705 

9-94  438 
9-94431 

37 
36 

12.0  II. 5  11,0 
14.4  13.8  13.2 
16.8  16.1  15.4 
19.2  18.4  17.6 
21.6  20.7  19.8 

25 

9.67  750 

24 

9-73326 

0.26  674 

9.94  424 

35 

26 

9-67  773 

23 

9-73356 

0.26644 

9.94417 

34 

9 

27 

9-67  796 

23 

9-73  386 

0.26614 

9.94410 

33 

28 

9.67  820 

24 

9.73416 

0.26  584 

9.94  404 

32 

29 
30 

31 

9.67  843 

23 
23 
24 

9-73  446 

30 
30 
31 
30 
30 
30 
30 
30 
30 

0.26  554 

9-94  397 

31 
30 

29 

7    6 

9.67  866 

9.73476 

0.26  524 

9-94  390 

9.67  890 

9-73507 

0.26493 

9-94  383 

32 

9.67913 

23 

9-73  537 

0.26  463 

9-94  376 

28 

0.7   0.6 
1.4   1.2 

33 

9-67  936 

23 
23 

9-73567 

0.26433 

9-94  369 

27 

2 

34 

9-67  959 

9.73597 

0.26403 

9-94  362 

26 

3 

2.1   1.8 

3S 

9.67  982 

^3 

9.73627 

0.26373 

9-94  355 

25 

4 

2.8   2.4  ' 

36 

9.68  006 

24 

9-73657 

0.26  343 

9.94  349 

24 

5 

3-5   3.0 

37 

9.68029 

23 

9.73  687 

0.26313 

9-94  342 

23 

6 

4.2   3.6 

3^ 

9.68052 

23 

9.73717 

30 
30 
30 
30 

0.26  283 

9-94  335 

22 

7 

4.9   4-2 

39 
40 

41 

9.68075 

^3 
23 
23 

9-73  747 

0.26  253 

9.94328 

21 
20 

19 

8 
9 

5-6   4.8 
6-3   5-4 

9.68  098 

9.73777 

0.26  223 

9.94321 

9.68  121 

9.73  807 

0.26  193 

9-94314 

42 

9.68  144 

23 

9-73  837 

30 
30 

0.26  163 

994307 

18 

43 

9.68  167 

23 

9.73  867 

0.26  133 

9-94  300 

17 

44 
46 

9.68  190 
9.68213 
9.68  237 

23 
23 
24 

9.73  897 
9.73927 
9-73  957 

30 
30 
30 
30 

0.26  103 
0.26073 
0.26  043 

9.94  293 
9.94  286 

16 
15. 

9-94  279 

V 

14 

47 

9.68  260 

23 

9-73987 

0.26013 

9-94  273 

13 

48 
49 
50 

SI 

9.68  283 
9-68  305 

23 

22 

23 
23 

9-74017 
9.74047 

30 
30 
30 
30 
30 
29 
30 

0.25  983 
0.25  953 

9.94  266 
9-94259 

12 
II 
(0 

9 

7   6   6 
31   31   30 

9.68  328 

9.74077 

0.25  923 

9-94  252 

0 

I 

2.2  2.6  2.5 

6.6  7.8  7-5 

II. I  12.9  12.5 

15.5  18.1  17.5 

19.9  23.2  22.5 

9.68351 

9.74  107 

0.25  893 

9-94  245 

52 
53 
54 

9-68  374 
9-68  397 
9.68  420 

23 
23 

23 

9-74  137 
9.74  166 

9-74  196 

0.25  863 
0.25  834 
0.25  804 

9-94  238 
9-94  231 
9.94  224 

8 
7 
6 

2 
3 
4 

55 

9.68443 

23 

9.74  226 

30 
30 

0.25  774 

9-94217 

5 

5 
6 

24.4  28.4  27.5 

5^ 

9.68466 

9-74  256 

0.25  744 

9.94  210 

4 

28.8  —  — 

57 

9.68489 

^3 

9.74  286 

0.25  714 

9.94  203 

3 

y 

5« 
59 
60 

9.68512 
9-68  534 

23 
22 

23 

9.74316 
9.74  345 

29 
30 

0.25  684 
0.25  655 

9.94  196 
9-94  189 

2 
I 

0 

. 

968557 

9-74  375 

0.25  625 

9.94  182 

L.  Cos.  1 

d. 

L.  Cot.  |c.  d. 

L.Tan.  j  L.  Sin. 

d.   '  1 

p.p. 

/»lo 


55 


/ 

L.  Sin. 

d. 

L.  Tan. 

cdT 

L.  Cot. 

L.  Cos. 

d.  in 

P.P. 

o 

9.6S  557 

23 

23 

9.74  375 

30 
30 
30 
29 

30 
30 
29 

30 
30 
30 
29 
30 
30 
29 
30 
30 
29 
30 
29 

30 
29 
30 
30 
29 
30 
29 
30 
29 
30 
29 
30 
29 
30 
29 
29 
30 
29 
30 
29 
29 
30 
29 

0.25  625 

9.94  182 

60 

I 

9.08  5S0 

9.74  405 

0.25  595 

9-94175 

59 

2 

9.68  603 

9.74  43? 

0.25  565 

9.94  168 

58 

3 

9.68  625 

23 
23 
23 

22 

23 
23 
22 

23 
22 
23 
23 

9.74465 

0-25  535 

9.94  1 61 

57 

30  29  23 

4 

9.68  648 

9.74  494 

0.25  506 

9-94  154 

56 

f 

3.0  2.9  2.3 

6.0  5.8  4.6 

9.0  .8.7  6.9 

12.0  1 1.6  9.2 

5 
6 

7 

9.68671 
9.68  694 
9.68  716 

9.74  524 
9-74  554 
9-74  583 

0.25  476 
0.25  446 
0.25417 

9-94  147 
9.94  140 

9-94  133 

55 
54 

53 

2 
3 
4 

8 

9.68  739 

9.74613 

0.25  387 

9.94126 

52 

s 

15.0  14.5  II. 5 

9 
10 

u 

9.68  762 

9-74  643 

0-25  357 

9.94  119 

51 
50 

49 

6 

7 
8 

18.0  17.4  13.8 
21.0  20.3  16.1 
24.0  23.2  18.4 

9.68  784 

9.74  673 

0.25  327 

9.94  "2 

9.68  807 

9.74  702 

0.25  298 

9.94  105 

12 

9.68  829 

9-74  732 

0.25  268 

9-94  098 

48 

9 

27.0  26.1  20.7 

13 

9.68852 

9.74  762 

0.25  238 

9-94  090 

47 

14 

9.68875 

9.74  791 

0.25  209 

9.94  083 

46 

IS 

9.68  897 

23 
22 

23 

22 

23 

22 

23 
22 

9.74821 

0.25179 

9-94  076 

45 

1 

i6 
17 
18 
19 
20 
21 

9.68  920 
9.68  942 
9-68  965 
9.68  987 

9.74851 

9.74  880 
9.74910 
9-74  939 

0.25  149 
0.25  120 
0.25  090 
0.25  061 

9.94  069 
9.94  062 
9-94055 
9.94  048 

44 

43 
42 

41 

40 

39 

I 

2 
3 

4 

S 

22  8   7 

2.2  0.8  0.7 

4.4  1.6  1.4 

6.6  2.4  2.1 

8.8  3.2  2.8 

ii.o  4.0  3.5 

9.69010 

9.74  969 

0.25  031 

9.94  041 

9.69  032 

9.74  998 

0.25  002 

9-94  034 

22 

9.69  055 

9.75  028 

0.24  972 

9.94  027 

3>^ 

6 

13.2  4.8  4.2 

23 

9.69  077 

23 

9.75  058 

0.24  942 

9.94  020 

37 

7 

15-4  5-6  4-9 

24 

9.69  100 

9.75087 

0.24913 

9.94012 

36 

8 

17.6  6.4  5.6 

2S 

9.69  122 

9-75117 

0.24  883 

9-94  005 

35 

9 

19.8  7.2  6.3 

26 

9.69  144 

23 

9.75  146 

0.24  854 

9-93  998 

34 

27 

9.69  167 

9.75  176 

0.24  824 

9-93991 

33 

28 

9.69  189 

23 

22 

22 
23 

9-75  205 

0.24  795 

9.93  984 

32 

29 

30 

31 

9.69  212 

9-75  23I 

0.24  765 

9-93  977 

31 
30 

29 

9.69  234 

9.75  264 

0.24  736 

993970 

9.69  256 

9-75  294 

0.24  706 

9-93  9^3 

32 

9.69  279 

9.75  323 

0.24677 

9-93  955 

28 

33 

9.69  301 

22 

9-75  353 

0.24  647 

9-93  948 

27 

34 

9-69  323 

"7 

9.75  382 

0.24  618 

9-93  941 

26 

8    8 

3S 

9-69  345 

23 
22 
22 

9-75  411 

0.24  589 

9-93  934 

25 

30   29 

3^ 
37 

9.69  368 
9.69  390 

9-75  441 
9-75  470 

0.24559 
0.24  530 

9.93  927 
9.93  920 

24 
23 

0 

I 

1.9   1.8 
5.6   5.4 

3« 

9.69412 

9.75  500 

0.24  500 

9.93912 

22 

2 

9.4   9.1 

39 
40 

41 

9.69434 

22 
23 

9-75  529 

0.24471 

9-93  905 

21 
20 

19 

3 
4 

13.1  12.7 
16.9  16.3 
20.6  19.9 

9.69  456 

975558 

0.24  442 

9.93  898 

9.69479 

9.75  588 

0.24412 

9-93  891 

42 

9.69  501 

22 

9.75617 

0.24  z^z 

9-93  884 

18 

7 
8 

24-4  23.6 

43 

9.69523 

9-75  647 

29 
29 
30 
29 
29 
29 
30 
29 

29 
29 

30 
29 
29 
29 

30. 
29 

29 

0.24  353 

9-93  876 

17 

28.1  27.2 

44 

9-69  545 

22 

9.75  676 

0.24  324 

9.93  869 

16 

4S 

9.69  567 

'7'? 

9-75  705 

0.24  295 

9-93  862 

IS 

46 

9.69  589 

9-75  735 

0.24  265 

9-93  855 

14 

47 

9.69  611 

O'J 

9-75  764 

0.24  236 

9-93  847 

13 

48 

9.69  633 

9-75  793 

0.24  207 

9.93  840 

12 

7    7 

49 
50 

51 

S2 

9.69  655 

22 

22 
22 
2'' 

9.75  822 

0.24  178 

9-93  833 

II 

10 

9 
8 

30   29 

9.69  677 

9.75  852 

0.24  148 

9.93  826 

0 

I 
2 

2.1   2.1 

6.4   6.2 
10.7  10.4 

9.69  699 
9.69  721 

9.75  881 
9.75910 

0.24  119 
0.24  090 

9.93819 
9-93811 

53 
S4 

9-69  743 
9.69  765 

22 
00 

9-75  939 
9-75  969 

0.24061 
0.24031 

9.93  804 
9-93  797 

7 
6 

3 
4 
5 
6 

15.0  14.5 
19.3  18.6 

ss 

9.69  787 

9.75  998 

0.24  002 

9-93  789 

S 

23.6  22.8 

56 

9.69  809 

22 

9.76  027 

0.23  973 

9.93  782 

4 

27.9  26.9 

S7 

9.69831 

9.76056 

0.23  944 

9.93  775 

3 

/  ' 

S8 

9.69  853 

22 

9.76  086 

0.23914 

9.93  768 

2 

4^ 
60 

9.69  875 

22 

9.76  115 

0.23  885 

9.93  760 

I 
0 

9.69  897 

9.76  144 

0.23  856 

9.93  753 

1  L.  Cos. 

d. 

L.  Cot. 

c.  d. 

L.  Tan. 

L.  Sin. 

d. 

' 

P.P. 

Qn° 


56 


L.  Sin.  I  d.  I  L.  Tan. 


c.  d, 


L.  Cot. 


L.  Cos. 


P.  P. 


9.69  897 


2 
3 
4 

5 
6 

7 
8 

9 

10 

II 
12 
13 
14 
15 
16 

17 
18 

19 

20 

21 
22 
23 
24 
25 
26 

27 
28 
29 

30 

31 

32 
33 
34 
35 
36 

37 
38 
39 
40 

41 

42 

43 
44 
45 
46 

47 
48 

49 

50 

51 

52 
53 
54 
55 
56 

57 
58 
59 
60 


9.69919 
9.69  941 
9.69  963 

9.69  984 

9.70  cx)6 
9.70028 
9.70  050 
9.70072 
9.70093 


.70115 


9.70137 
9.70159 
9.70  180 
9.70  202 
9.70  224 
9.70  245 
9.70  267 
9.70  288 
9.70310 


9.70  332 


9-70353 
970  37^ 
9.70  396 

9.70418 

9-70  439 
9.70461 

9.70482 
9.70  504 
9.70  525 


970547 


9.70  568 
9.70  590 
9.70  611 

970  633 
9.70  654 
9.70  675 
9.70  697 
9.70718 
970  739 


9.70  761 


9.70  782 
9.70  803 
9.70  824 
9.70846 
9.70  867 
9.70  888 
9.70  909 
9.70931 
9.70952 


970973 


9.70  994 

9.71  015 
9.71  036 
9.71058 
9.71  079 
9.71  100 
9.71  121 
9.71  142 
9.71  163 


971  184 


9.76  144 


9.76  173 
9.76  202 
9.76231 
9.76  261 
9.76  290 
9.76319 
9.76348 

976377 
9.76406 


976435 
976  464 
976  493 
9.76  522 

976551 
9.76  580 
9.76  609 
9.76639 
9.76  668 
9.76697 


9.76  725 


976  754 
9.76  783 
9.76812 

9.76841 
9.76870 
9.76899 
9.76928 

976957 
9.76986 


977o»g 
977  044 
977073 
9.77  lOI 

977  130 

977  159 
977  188 
9.77217 

977  246 
9.77274 


977  303 


977  332 
9.77361 
977  390 
9.77418 

977  447 
9.77476 

977  505 
977  533 
9.77562 


977591 


9.77619 
977  648 
9.77677 

977  706 
977  734 
977  763 

977  791 
9.77820 

977  849 


9.77877 


29 
29 
29 
30. 
29 
29 
29 
29 

29 
29 

29 
29 
29 
29 
29 
29 
30 

29 
29 
28 

29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
28 
29 
29 
29 
29 
29 
28 
29 
29 
29 
29 
28 

29 
29 
29 
28 
29 
29 
28 

29 
29 
29 
28 
29 
28 

29 
29 
28 


0-23  856 
0.23  827 
0.23  798 
0.23  769 

0.23  739 
0,23  710 
0.23  681 
0.23  652 
0.23  623 
023594^ 

0-23  565 


9-93  753 


9-93  746 
9-93  738 
9-93  731 
9-93  724 
9-93717 
9-93  709 
9.93  702 

9-93  695 
9.93  687 


9-93  680 


0.23  536 
0.23  507 
0.23  478 
0.23  449 
0.23  420 
0.23  39  f 
0.23  361 
0.23  332 
0.23  303 


0.23  275 


0.23  246 
0.23  217 
0.23  188 
0.23  159 
0.23  130 
0.23  lOI 
0.23072 
0.23  043 
0.23014 


0.22  985 


0.22956 
0.22927 
0.22  899 
0.22  870 
0.22  841 
0.22  812 
0.22  783 
0.22  754 
0.22  726 


0.22  697 


0.22  668 
0.22  639 
0.22  610 
0.22  582 
0.22553 
0.22  524 
0.22495 
0.22467 
0.22  438 


0.22  409 


0.22  381 
0.22  352 
0.22  323 
0.22  294 
0.22  266 
0.22  237 
0.22  209 
0.22  180 
0.22  151 
0.22  123 


9-93  673 
9.93  665 

9-93  658 

9.93  650 

9-93  643 
9-95636 

9.93  628 
9.93621 
9.93614 


9.93  606 


9-93  599 
9.93591 
9-93  584 

9-93  577 
9-93  569 
9.93  562 

9-93  554 
9-93  547 
9-93  539 


9-93  532 


9-93  525 
9-93517 
9-93510 
9.93  502 
9-93  495 
9-93  487 
9.93  480 
9-93  472 
9-93  465 


-93  457 


9-93  450 
9-93  442 
9-93  435 
9-93427 
9-93  420 
9.93412 

9-93  405 
9-93  397 
9-93  390 


9-93  382 


9-93  375 
9-93  367 
9-93  360 
9-93352 
9-93  344 
9-93  337 

9-93  329 
9.93  322 

9-93  3H 

9-93  307 


I  L.  Cos.  I  d.  I  L.  Cot.  Ic.  d.|  L.  Tan.  |  L.  Sin. 


60 

59 
58 
57 
56 
55 
54 

53 
52 
51 
50 

49 
48 
47 
46 
45 
44 

43 
42 
41 
40 

39 
38 
37 
36 
35 
34 

33 
32 
31 
30 

29 
28 
27 
26 
25 
24 

23 
22 
21 

20 

19 
18 

17 
16 
15 
14 

13 
12 
II 

10 

9 
8 

7 
6 
5 
4 

3 
2 
i 


30  29  28 


30 

6.0 

9.0 

12.0 

15.0 


2.9 

5.8 

8.7 

11.6 

14-5 
18.0  17.4 
21.0  20.3 
24,0  23.2  22.4 
27.0  26.1  25.2 


2.8 
5-6 
8.4 
II. 2 
14.0 
16.8 
19.6 


22   21 


2.2 

4-4 
6.6 
8.8 

II.O 

13.2 
15.4 
17.6 
19.8 


0.8 
1.6 
2.4 

3-2 

4.0 

4.8 

5-6 
6-4 

7-2 


2.1 

4.2 

6.3 

8.4 

10.5 

12.6 

14.7 
16.8 
18.9 


0.7 

1-4 
2.1 

2.8 

3-5 
4.2 

4.9 
5-6 
6.3 


30  29  28 

2.1  2.1  2.0 

6.4  6.2  6.0 

10.7  10.4  1 0.0 

15.0  14.5  14.0 

19.3  18.6  18.0 

23.6  22.8  22.0 

27.9  26.9  26.0 


P.  P. 


57 

1 

L.  Sin. 

["d. 

L.  Tan. 

c.  d. 

L.  Cot. 

L.  Cos. 

d. 

P.  P. 

0 

I 

9.71  184 
9.71  205 

21 
0 1 

9.77877 
977  906 

29 
29 
28 

0.22  123 

9-93  307 

8 

60 

59 

0.22  094 

9.93  299 

2 

9.71  226 

2 1 

977  935 

0.22  ob5 

9.93291 

7 
8 

7 
8 

58 

3 

9.71  247 

977  9t>3 

29 

28 
29 
'>8 

0.22037 

9.93  284 

57 

4 

9.71  268 

21 

9.77992 

0.22008 

9.93  276 

56 

29       28 

5 

9.71  289 

9.78020 

0.21  980 

9.93  2b9 

55 

I 

2.9       2.8 

6 

9.71  310 

9.78049 

0.21  951 

9.93  2bl 

8 

54 

2 

5.8      5.6 

7 

9-71  331 

21 
21 
20 
21 
21 
21 
21 

9.78077 

29 
29 
28 

29 
28 
29 
28 

29 

98 

0.21  923 

9-93  253 

7 
8 
8 

7 
8 
8 

7 
8 

53 

3 

8.7       8-4 

8 

971  352 

9.78  lob 

0.21  894 

9.93  24b 

52 

4 

11.6     II. 2 

9 
10 

II 

971 IIZ 

978  135 

0.21  8b5 

9-93  238 

51 
50 

49 

I 

7 
8 

9 

14.5     14.0 
17.4     16.8 
20.3     19.6 

971  393 
9.71  414 

9.78  163 

0.21  837 

9-93  230 

9.78  192 

0.21  808 

9-93  223 

12 

13 

971  435 
971456 

9.78  220 
9.78  249 

0.21  780 
0.21  751 

9-93215 
9.93  207 

48 
47 

23.2     22.4 
26.1     25.2 

14 

9.71  477 

9.78277 

0.21  723 

9.93  200 

46 

15 

9.71498 

'>  I 

9.78  30b 

0.21  b94 

9.93  192 

c 

45 

lb 

971  519 

978  334 

0.2 1  (^iib 

9.93  184 

44 

17 

971  539 

20 

978  2>^Z 

29 

2S 

0.21  b37 

9-93177 

7 
8 

43 

21        20 

18 

9.71  5bo 

21 

9.78391 

28 

0.21  bo9 

9.93  ib9 

8 

42 

I 

2.1       2,0 

19 
20 

21 

9-^  581 

21 

20 

978419 

29 
28 

29 
28 

0.21  581 

9.93  ibi 

7 
8 
8 

7 
8 

8 

41 
40 

39 

2 

3 
4 

4.2       4.0 
b.3       bo 
8.4       8.0 

9.71  602 

978448 

0.21  552 

9-93154 

9.71  b22 

9.7847b 

o.5i  524 

9.93  146 

22 

971  643 

978  505 

0.21  495 

9.93  138 

:<^ 

5 

10.5      lO.O 

23 

9.71  bb4 

978533 

0.21  4b7 

9-93131 

31 

b 

12. b     12.0 

24 

971  685 

20 

9.78  5b2 

29 

0% 

0.21  438 

9.93  123 

36 

I 

14.7     140 
ib.8     ibo 

2S 

9.71  705 

9.78  590 

28 

29 

28 

0.21  410 

9-93115 

35 

9 

18.9     18.0 

26 
27 

9.71  72b 
9.71  747 

21 

9.78  bi8 
9.78  b47 

0.21  382 
0.21  353 

9.93  108 
9.93  100 

7 
8 
8 

34 
2,3 

28 

9.71  7b7 

■7  T 

978675 

29 
28 

28 
29 
28 

0.21  325 

9.93  092 

8 
7 
8 

8 
8 

32 

29 

30 

31 

971  788 

21 

20 

9.78  704 

0.21  29b 

9.93  084 

31 
30 

29 

8        7 

9.71  809 

9.78  732 

0.21  2b8 

9-93077 

9.71  829 

9.78  7bo 

0.21  240 

993069 

I 

i  0.8     0.7 

32 

9.71  850 

20 

9.78  789 

0.21  211 

9-93061 

28 

2 

|i.6  I.; 

33 

9.71  870 

21 

978817 

28 

0.21  183 

993053 

7 
8 
8 

27 

3 

1 2.4  2.1 

34 

9.71  891 

-70 

9.78  845 

29 
28 

0.21  155 

9-93  046 

26 

4 

13-2  2.8 

35 

9.71  911 

9.78  874 

0.21  12b 

9-93  038 

25 

5 

4.0    3-5 

37 

9.71  932 
9.71952 

20 

9.78  902 
9.78930 

28 

29 

28 

0.21  098 
0.21  070 

9-93  030 
9.93022 

8 
8 

24 
23 

I 

4.8    4.2 
5-6    4-9 
6.4     5.6 

3« 

971  973 

21 

978  959 

0.21  041 

9-93014 

7 
8 

8 

8 

22 

39 
40 

41 

9.71  994 

20 
20 
21 

978  987 

28 
28 
29 
28 
28 

0.21  013 

9-93  007 

21 
20 

19 

9  1  /"^     ^-j 

9.72014 

9.79015 

0.20  985 

9-92  999 

9.72034 

979043 

0.20957 

9.92991 

42 

43 

9.72055 
9.72075 

20 
21 

9.79072 
9.79  100 

0.20  928 
0.20  900 

9-92  983 
9.92976 

7 
8 

18 
17 

44 

9.72  096 

20 

9.79  128 

28 

0.20872 

9.92  968 

8 

16 

45 

9.72  lib 

979  156 

29 

0.20844 

9.92  9bo 

8 

15 

4b 

9.72  137 

20 

979  185 

0.20  815 

9.92952 

8 

14 

8         8         8 

47 

9.72157 

20 

9.79213 

-'S 

0.20  787  • 

9.92  944 

8 

13 

30      29       28 

48 

9.72177 

9.79241 

28 

0.20  759 

9-92936 

7 
8 

8 

8 

12 

0 

49 
50 

51 

9.72  198 

20 
20 
21 

979  269 

28 
29 

0.20731 

9.92  929 

II 
10 

9 

I 
2 
3 

1.9      1.8      1.8 

5-6     5'4     5-2 

9.4     9.1      8.8 

13.1    12.7    12.2 

ib.9    ib.3    15.8 

20.b    19.9    19.2 

9.72  218 

979  297 

0.20  703 

9.92921 

9.72238 

979  326 

0.20  b74 

9.92913 

52 

9.72259 

20 

979  354 

->« 

0.20  b4b 

9.92  905 

s 

8 

4 

53 

9.72  279 

20 

979  382 

?8 

0.20  bi 8 

9-92  897 

8 

7 

5 
b 

54 

9.72  299 

21 

9.79410 

28 

0.20  590 

9.92  889 

8 

b 

24.4    23.b    22.8 

55 

9.72  320 

20 

979438 

28 

0.20  5b2 

9.92  881 

7 
8 
8 

5 

.  8 

28.1    27.2    2b. 2 

5^^ 

57 

9.72  340 
9.72  3bo 

20 

979466 
979  495 

29 
28 

0.20  534 
0.20  505 

9.92  874 
9.92  8bb 

4 
3 

5« 

9.72381 

20 

979523 

->« 

0.20477 

9.92  858 

s 

2 

59 
60 

9.72401 

20 

979551 

28 

0.20  449 

9  92  850 

8 

0 

972421 

979  579 

0.20421 

9.92  842 

L.  Cos. 

d. 

L.  Cot. 

c.  d. 

L.  Tan. 

L.  Sin. 

d. 

/ 

P.P. 

.^»^ 


58 


/ 

L.  Sin. 

d. 

L.  Tan. 

0.  d. 

L.  Cot. 

L.  Cos. 

d. 

P.P. 

0 

I 

9.72421 

20 

979  579 

28 
28 
28 

0.20421 

9.92  842 

8 
8 

60 

59 

9.72441 

979  607 

0.20  393 

9.92  834 

2 

9.72461 

979  635 

0.20  365 

9.92  826 

s 

58 

3 

9.72  482 

20 
20 

979  663 

28 
28 

0.20  337 

9.92818 

8 

I 

57 

4 

9.72  502 

979  691 

0.20  309 

9.92  810 

56 

29     28     27 

5 

9.72522 

20 

9.79719 

28 

0.20  281 

9.92  803 

55 

I 

2.9     2.8     2.7 

6 

9.72542 

20 

20 

979  747 

29 
28 

0.20  253 

9-92  795 

« 

54 

2 

5.8     5.6     5.4 

7 

9.72562 

979  776 

0.20  224 

9.92  787 

8 

53 

3 

8.7     8.4     8.1  ■ 

8 

9.72582 

979  804 

0.20  196 

9.92  779 

52 

4 

11.6  11.2  10.8 

9 
10 

II 

9.72  602 

20 
21 
20 

9.79832 

28 
28 

0.20  168 

9.92  771 

8 
8 

8 

51 
50 

49 

5 
6 

14.5   14.0  13.5 
17.4  16.8  16.2 
20.3  19.6  18.9 
23.2  22.4  21.6 

9.72  622 

9.79  860 

0.20  140 

9.92  763 

9.72  643 

9.79  888 

0.20  112 

992  755 

12 

9.72  663 

20 

9.79916 

->« 

0.20084 

9.92  747 

8 

48 

9 

26.1    21;. 2    2^.1 

9.72  683 
9.72  703 

20 
20 

979  944 
979  972 

28 
28 

0.2005b 
0.20028 

992  739 
9.92  731 

8 
8 

47 
46 

15 

9.72  723 

9.80000 

28 

0.20000 

9.92  723 

8 

45 

16 

972  743 

9.80  028 

28 
28 
28 
28 
28 

0.19972 

9.92715 

8 
8 
8 
8 
8 
8 
8 
8 
8 

44 

17 

9.72  763 

20 

-20 

20 

20 

9.80  056 

0.19  944 

9.92  707 

43 

21     20     19 

iS 

9-72  783 

9.80  084 

0.19916 

9.92  699 

42 

I 

2.1       2.0      1.9 

19 
20 

21 

9.80  112 

0.19888 

9.92691 

41 
40 

39 

2 
3 
4 

4.2  4.0      3.8 

6.3  6.0      5.7 

8.4  8.0      7.6 

9.72  823 

9.80  140 

0.19  860 

9.92  683 

9.72  843 

9.80  168 

0.19832 

9.92  675 

22 
23 
24 

9.72863 
972  883 
9.72  902 

20 

19 
20 

9.80  195 
9.80  223 
9.80251 

27 
28 
28 
28 

0.19  805 
0.19777 
0.19  749 

9.92  667 
9.92  659 
9.92651 

38 
37 
36 

5 
6 

I 

10.5  lo.o    9.5 

12.6  12.0  H.4 

14.7  14.0  13.3 

16.8  16.0  15.2 

18.9  18.0  I7.I 

25 

9.72922 

9.80  279 

-pR 

0.19  721 

9.92  643 

8 
8 
8 
8 
8 
8 
8 
8 

35 

9 

26 

9.72942 

9.80  307 

28 

28 
08 

0.19693 

9-92  635 

34 

27 

9.72  962 

9-8o  335 

0.19  665 

9.92627 

33 

28 

9.72982 

9.80  363 

0.19637 

9.92  619 

32 

29 
30 

31 

9.73002 

20 
19 

9.80391 

28 
28 
27 
28 

0.19609 

9.92  611 

31 
30 

29 

9       8       7 

9.73022 

9.80419 

0.19581 

9.92  603 

9.73041 

9.80447 

0.19553 

9.92  595 

T 

0.9    0.8    0.7 

32 

9.73061 

9.80  474 

0.19  526 

9.92  587 

28 

2 

1.8    1.6    1.4 

33 
34 

9.73081 

9.73  lOI 

20 
26 
19 

9.80  502 
9.80  530 

28 
28 
28 

0.19498 
0.19470 

9-92  579 
9.92571 

8 
8 
8 

27 
26 

3 

4 

2.7    2.4    2.1 
3.6    3.2    2.8 

35 

9.73  121 

9.80558 

0.19442 

9.92  563 

25 

5 

4-5    4.0    3.5 

3^ 

973  140 

9.80  586 

28 
28 

0.19  414 

9-92  555 

24 

6 

5.4    4.8    4.2 

37 

973  160 

9.80614 

0.19  386 

9.92  546 

9 
8 
8 
8 
8 
8 
8 

23 

7 

6.3    5.6    4.9 

38 

973  180 

9.80  642 

0.19358 

9.92538 

22 

8 

7.2    6.4    5.6 

39 
40 

41 

973  200 

19 
20 
20 
19 

9.80  669 

2y 
28 

28 
28 
28 

0.19  331 

9.92  530 

21 
20 

19 

9 

8.1    7.2    6.3 

9.73219 

9.80  697 

0.19  303 

9.92  522 

973  239 

9.80  725 

0.19275 

9.92514 

42 
43 
44 

973259 
9.73278 

973  298 

9-8o  753 
9.80  781 

9.80  808 

0.19  247 
0.19  219 
0,19  192 

9.92  506 
9.92  498 
9.92  490 

18 
17 
16 

20 

27 
28 
28 

8 
8 
9 
8 
8 

45 

973318 

19 

9.80  836 

0.19  164 

9.92482 

15 

4b 

973  337 

9.80  864 

28 
27 

0.19  136 

9.92473 

14 

47 

973  357 

9.80  892 

0.19  108 

9.92  465 

13 

8        8        7 

48 

973  377 

19 
20 

9.80919 

0.19  081 

9.92457 

s 

12 

sa     2S      28 

49. 

973  396 

9.80  947 

-^8 

0.19053- 

_2:S2^9 

'8- 

11 

0 

50 

51 

52 
53 
54 
55 

9.73416 

19 
20 
19 
20 

19 
20 

9.80975 

28 

'27 
28 
28 

0.19025 

9.92441 

8 
8 
9 
8 
8 
8 

10 

9 
8 

7 
6 

5 

I 

2 

3 
4 

1.8     1.8     2.0 
5.4     5.2     6.0 
9.1     8.8  lo.o 
12.7  12.2  14,0 
16.3  15.8  18.0 
19.9  19.2  22.0 
23.6  22.8  26.0 

973  435 
973  455 
973  474 

973  494 
973  5'3 

9.81  003 
9.81  030 
9.81  058 
9.81  086 
9.81  113 

0.18997 
0.18970 
0.18942 
0.18914 
0.18  8S7 

992433 
9.92425 
9.92416 
9.92  408 
9.92  400 

5^ 

973  533 

19 

9.81  141 

28 

0.18859 

9.92  392 

8 

4 

8 

27.2  26.2    — 

57 

973552 

9.81  169 

0.18  831 

9.92  384 

8 

9 
8 

3 

58 

973572 

19 
20 

9.81  196 

27 

0.18804 

9.92376 

2 

59 
60 

973591 

9.81  224 

28 

0.18  776 

9.92  367 

I 
0 

973  611 

9.81  252 

0.18748 

992  359 

L.  Cos. 

d. 

L.  Cot. 

c.  d. 

L.  Tan. 

L.  Sin. 

d. 

/ 

P.P. 

K7' 


00 

.^9 

/ 

L.  Sin. 

d. 

L.  Tan. 

c.  d. 

L.  Cot. 

L.  Cos. 

d. 

P.P. 

0 

I 

9.73611 

19 

9.81  252 

27 
28 

0.18748 

9.92  359 

8 

8 

60 

59 

9-73  630 

9.81  279 

0.18  721 

9.92351 

2 

9.73  650 

9.81  307 

28 
27 
28 

0.18693 

9.92  343 

s 

58 

3 
4 

9.73  669 
9.73689 

19 
20 

19 
19 
20 

19 

9.81  33-5 
9.81  362 

0.18665 
0.18638 

9.92  33^ 
9.92  326 

9 

s 

57 
56 

28   27 

S 

9.73  708 

9-81  390 

28 

0.18  610 

9.92318 

8 

55 

J  ,   ,, 

2.^    2.7 

6 
7 

9.73727 
9-73  747 

9.81  418 
9.81  445 

27 
28 

0.18582 
0.18555 

9.92310 
9.92  302 

8 

9 

X 

54 
53 

2 
3 
4 

5-6   5-4 
8.4   8.1 

U.2   10.8 

8 

9.73  766 

9.81  473 

27 
28 
28 

11 

0.18  527 

9.92  293 

S2 

9 
iO 

II 

9-73  785 

19 
20 

19 

9.81  500 

0.18500 

9-92  285 

8 
8 
9 

s 

51 
50 

49 

1 

14.0   13.5 
16.8  l6.2 

19.6  18.9 

22.4  21.6 

9.73  «o5 

9.81  528 

0.18472 

9-92  277 

9.73824 

9.81  556 

0. 1 8  444 

9.92  269 

12 

9.73  843 

19 

9.81  583 

0.18417 

9.92  260 

48 

9 

25.2  24.3 

13 

9-73  863 

19 

19 

9.81  611 

27 
28 

0.18389 

9.92252 

ft 

47 

14 

9.73882 

9.81  638 

0.18362 

9.92  244 

9 
8 

46 

'5 

9.73901 

9.81  666 

27 

28 

0.18334 

9.92  235 

45 

i6 

9.73921 

19 

9.81  693 

0.18307 

9.92227 

8 

44 

20  19  18 

»7 

9.73  940 

19 
19 
19 
20 

9.81  721 

27 
28 
27 
28 

0.18  279 

9.92219 

8 

43 

i8 

9.73959 

9.81  748 

0.18  252 

9.92  211 

9 
8 

8 

42 

I 

2.0  1.9  1.8 

19 
20 

21 

9-73978 

9.81  776 

0.18224 

9.92  202 

41 
40 

39 

2 
3 
4 

4.0  3-8  3.6 
6.0  5.7  5.4 
8.0  7.6  7.2 

9-73  997 

9.81  803 

0.18  197 

9.92  194 

9.74017 

9.81  831 

0.18  169 

9.92  186 

22 
23 
24 

9.74036 
9-74055 
9.74074 

^9 
19 
19 
19 

9.81  858 
9.81  886 
9.81  913 

28 
27 

0.18  142 
0.18  114 
0.18087 

9.92  177 
9.92  169 
9.92  161 

9 
8 

8 

9 

8 

38 
37 
36 

'e 

lo.o  9.5  9.0 
12.0  1 1.4  10.8 
14.0  13.3  12.6 
16.0  15.2  14.4 

25 

9.74  093 

9.81  941 

27 
28 

0.18059 

9.92  152 

35 

9 

18.0  I7.I  16.2 

26 

9-74113 

19 
19 
19 
19 
19 
19 
19 

9.81  968 

0.18032 

9.92  144 

8 

34 

27 
28 

9-74  132 
9-74151 

9.81  996 

9.82  023 

% 

0.18004 
0.17977 

9.2.2  136 
9.92127 

9 

8 

33 
32 

29 

30 

31 

9.74170 

9.82051 

27 
28 
27 
28 

0.17949 

9.92  119 

8 

9 
8 
8 

31 
30 

29 

9    8 

9.74  189 

9.82  078 

0.17  922 

9.92  III 

9.74  208 

9.82  106 

0.17894 

9.92  102 

I 

0.9  0.8 

32 

9.74227 

9.82  133 

0.17867 

9.92  094 

28 

2 

1.8   1.6 

33 

9.74  246 

9.82  161 

0.17839 

9.92086 

27 

3 

2.7   2.4 

34 

9.74  265 

19- 

9.82  188 

2/ 

0.17  812 

9.92077 

9 
8 

26 

4 

3.6  3-2 

35 

9.74  284 

19 
19 
19 
19 

9.82  215 

28 
27 
28 

0.17785 

9.92  069 

25 

b 
6 

7 
8 

4.5  4.0 
5.4  4.8 

6.3   5-6 
7.2   6.4 
8.1   7.2 

3^ 

9.74  303 
9.74322 

9-82  243 
9.82  270 

0.17757 
0.17  730 

9.92  060 
9.92052 

9 
8 
ft 

24 
23 

3^ 

9-74  341 

9.82  298 

0.17702 

9.92  044 

22 

9 

39 
40 

41 

9-74  360 

19 
19 
19 
19 
19 

9.82325 

27 
27 
28 
27 
28 

0.17675 

9.92035 

9 
8 

9 
8 

21 
20 

19 

9.74379 

9.82352 

0.17648 

9.92027 

9.74  398 

9.82  380 

0.17  620 

9.92018 

42 

9.74417 

9.82  407 

0-17593 

9.92010 

8 

18 

43 
44 

9-74436 
9-74  45? 

9-82435 
9.82  462 

0.17565 
0-17538 

9.92002 
9.91  993 

17 
16 

19 
19 
19 
19 
19 
18 

27 

11 

27 

9 
8 
9 
8 

9 

8 

45 

9-74  474 

9.82  489 

0.17511 

9.91  985 

15 

4b 

47 
48 

9.74493 
974512 
9-74531 

9-82517 
9-82  544 
9-82571 

0.17483 
0.17456 
0.17429 

9.91  976 
9.91  968 
991  959 

14 

13 
12 

0 

9   9   8 
28   27   27 

49 
50 

51 

9-74  549 

19 
19 

9.82  599 

27 
27 
28 

0.17  401 

9.91  951 

9 

8 

II 
10 

9 

I 
2 

1.6  1.5  1.7 

4-7  4.5  5-1 

7.8  7.5  8.4 

10.9  10.5  1 1.8 

9.74  568 

9.82  626 

0.17374 

9.91  942 

9.74  587 

9.82653 

0.17347 

9-91  934 

S2 

9.74  606 

19 

9.82681 

0.17  319 

9.91  925 

9 
8 

9 
8 

8 

4 

53 
54 

9.74625 
9.74  644 

19 
19 
18 

9.82  708 
9-82  735 

2'J 
27 
27 
28 
27 

0.17292 
0.17  265 

9.91917 
9.91  908 

7 
6 

14.0  13.5  15.2 

17.1  16.5  18.6 

20.2  19.5  21.9 

55 
56 

9.74662 
9.74681 

19 
19 

9.82  762 
9.82  790 

0.17238 
0.17  210 

9.91  900 
9.91  891 

9 
8 

5 
4 

I 

23.3  22.5  25.3 

26.4  25.5  — 

57 

9.74  700 

9.82817 

0.17  183 

9.91  883 

3 

9 

5« 

9.74719 

19 

9.82  844 

27 
27 
28 

0.17  156 

9.91  874 

9 
8 

9 

2 

59 
60 

9-74  737 

19 

9.82871 

0.17  129 

9.91  866 

0 

9-74  756 

9.82  899 

0.17  lOI 

9.91  857 

L.  Cos. 

d. 

L.  Cot. 

c.  d.|  L.Tan. 

L.  Sin. 

d. 

' 

P.  P. 

6o 


L.  Sin. 


Tan. 


c.  d.l  L.  Cot.  I  L.  Cos. 


P.  P. 


O 

I 

2 

3 

4 
5 
6 

7 
8 

9 

10 

II 

12 

13 

H 

15 
16 

17 

18 

19 
20 

21 

22 

23 
24 

25 
26 

27 
28 
29 
30 

31 

32 
33 
34 
35 
36 

37 
38 
39 
40 

41 
42 

43 

44 

45 
46 

47 
48 

49 

50 

51 

52 
53 
54 
55 
56 

57 
58 
59 
60 


9-74  756 


9-74  775 
9-74  794 
9.74812 

9.74831 
9.74  850 
9.74  868 
9.74887 
9.74  906 
9.74  924 


9-74  943 


9.74961 

9.74  980 
9-74  999 
9.75017 

9.75  036 
9.75  054 

9-75  073 
9.75091 
9.75110 


9.75  128 


9.75  147 
9-75  165 
9-75  184 
9.75  202 
9.75  221 
9-75  239 
9-75  258 
9.75  276 
9-75  294 


9.75313 


9-75  331 
9-75  350 
9.75  368 

9-75  386 
9-75  405 
9-75  423 
9-75  441 
9-75  459 
9-75  478 


9-75  496 
9.75514 
9.75  533 
9.75551 
9.75  569 
9.75  587 
9.75  605 
9.75  624 
9.75  642 
9.75  660 


9.75  678 


9.75  696 
9.75  714 
9.75  733 
9.75751 
9.75  769 
9.75  787 
9-75  805 
9.75  823 
9.75  841 


9.75  859 


9.82  899 


9.82  926 

9.82953 

9.82  980 

9.83  008 

9.83  035 
9.83  062 

9.83  089 
9.83  117 
9.83  144 


9.83  n^ 
9.83  198 
9.83  225 
9.83  252 
9.83  280 
9-83  307 
9.83  334 
9-83  361 
9.83  388 
9.83415 


9.83  442 


9.83  470 
983497 
9.83  524 

9.83551 
9.83  578 
9.83  605 
9.83  632 
9.83  659 

9.83  686 


9.83713 


9.83  740 
9.83  768 
9.83  795 
9.83  822 

9.83  849 
9.83  876 

9.83  903 
9.83  930 
9.83957 


9.83  984 


9.84  on 

9.84  038 
9.84  065 

9.84092 
9.84  119 
9.84  146 

9.84173 
9.84  200 
9.84  227 


9.84  254 


9.84  280 
9.84  307 
9.84  334 
9.84  361 
9.84  388 
9.84415 
9.84  442 
9.84  469 
9.84  496 


I  9.84  523 


27 
27 
27 
28 
27 
27 
27 
28 
27 
27 
27 

27 
27 

28 
27 
27 
27 
27 
27 
27 
28 

27 
27 

27 
27 
27 
27 
27 
27 
27 
27 
28 
27 

27 

27 
27 

27 

27 

27 
27 

27 

27 
27 

27 

27 
27 

27 

27 
27 
27 

26 

27 
27 

27 

27 
27 
27 

27 
27 

27 


0.17  lOI 

0.17074 
0.17047 
0.17  020 
0.16992 
0.16965 
0.16938 
0.16  911 
0.16883 
0.16856 


0.16  829 


0.16  802 
0.16775 
0.16  748 
0.16  720 
0.16  693 
0.16666 
0.16  639 
0.16  612 
0.16  58g 
0.16558 


0.16530 
0.16  503 
0.16476 
0.16  449 
0.16422 
0.16395 
0.16368 
0.16  341 
0.16  314 


0.16  287 


0.16260 
0.16  232 
0.16  205 
0.16  178 
0.16  151 
0.16  124 
0.16097 
0.16070 
o.  1 6  043 


0,16016 

0.15989 
0.15  962 

0.15935 
0.15  908 
0.15  881 
0.15  854 
0.15  827 
o.  1 5  800 
0-15  773 


0.15  746 
0.15  720 
0.15693 
o.  1 5  666 
0.15  639 
0.15  612 
0.15585 
0.15558 
0.15531 
0.15504 


9.91  857 


9.91  849 
9.91  840 
9.91  832 
9.91  823 
9.91  815 
9.91  806 
9.91  798 
9.91  789 
9.91  781 


9.91  772 


9.91  763 

9.91  755 
9.91  746 

9.91  738 
9.91  729 
9.91  720 
9.91  712 
9.91  703 
9.91  695 


9.91  686 


9.91  677 
9.91  669 
9.91  660 
9.91  651 
9.91  643 
9.91  634 
9.91  625 
9.91  617 
9.91  608 


9.91  599 


9.91  591 
9.91  582 
9.91  573 

9.91  565 
9.91556 
9.91  547 
9.91  538 
9.91  530 
9.91  521 


.9»  512 


9.91  504 

9.91  49l 
9.91  486 

9.91  477 
9.91  469 
9.91  460 

9-91451 
9.91  442 

9.91  433 


9.91425 


15477 


9.91  416 
9.91  407 
9.91  398 
9.91  389 
9.91  381 
9.91  372 

9.91  363 
9.91  354 
9.91  345 


9.91 


60 

59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 

43 

42 

41 
40 

39 
38 
37 
36 
35 
34 

33 


30 

29 
28 
27 
26 
25 
24 

23 
22 
21 

20 

19 
18 

17 
16 
15 
14 

13 
12 
II 

10 

9 
8 

7 
6 
5 
4 
3 


27  26 


2.8 

5.6 

8.4 

II. 2 

14.0 


2.7 

5-4 

8.1 

10.8 

13.5 

16.8  16.2  15.6 

19.6  18.9  18.2 

22.4  21.6  20.8 

25.2  24.3  23.4 


2.6 

7.8 
10.4 
13.0 


18 

1.8 
3-6 
5.4 
7.2 
9.0 
1 1.4  10.8 
13.3  12.6 


19 

1.9 
3.8 
5.7 
7.6 

9-5 


15.2 
17.1 


14.4 
16.2 


9   8 


0.9 
1.8 
2.7 
3-6 
4-5 
5.4 
6.3 
7.2 
8.1 


0.8 
1.6 
2.4 
3-2 
4.0 
4.8 
5.6 
6.4 
7.2 


28 


28 


1.8 

5.2 


1.6 

4.7 

7.8  8.» 

10.9  12.2 

14.0  15.8 

17.1  19.2 

20.2  22.8 

23.3  26.2 

26.4  


27 

1.7 
51 
8.4 
11.8 
15.2 
18.6 
21.9 
25.3 


L.  Cos. 


d.  j  L.  Cot.  !c.  d 


j-^n^ 


L.  Sin. 


P.  P. 


6i 


J^    L.  Sin. 

d. 

L.  Tan.  c.  d.  L.Cot. 

L.  Cos. 

d.i   ' 

P.P 

0 

I 

9-75  «59 

18 
18 

9.84523 
9.84  550 

27 
26 

015477 

9-91  336 

8 

9 
9 
9 
9 
9 

60 

27 

26 

9-75^77 

0.15450 

9-91  328 

2 

9-75  «95 

18 

9.84576 

27 
27 
27 
27 

0.15424 

9.91  319 

58 

I 

2.7 

2.6  ■ 

3 

9-75913 

18 

9.84  603 

0.15397 

9.91  310 

57 

2 

5-4 

5-2 

4 

9.75931 

iS 

9.84  630 

0.15370 

9.91  301 

56 

3 

8.1 

7.8 

5 

9-75  949 

18 

9.84657 

0.15343 

9.91  292 

55 

4 

10.8 

10.4 

6 

9-75  9^7 

18 
18 
18 
18 
18 
18 
18 
18 
18 

9.84  684 

0.15  316 

9.91  283 

54 

5 

i3S 

13.0 

7 

9.75  9S5 

9.84  711 

27 

0.15  289 

9.91  274 

9 

53 

^ 

16.2 
18.9 
21.6 

15.6 
18.2 
20.8 
23-4 

17 

8 

9 

10 

II 

9.76003 
9.76021 

9.84  738 
9l84_7^ 
9.84791 
9.84818 

27 
26 
27 

27 

0.15  262 
0.15  236 

9.91  266 
9.91  257 

9 
9 
9 
9 
9 

52 
51 
50 

49 

i 

9.76  039 

0.15  209 

9.91  248 

9  1  ^^-j 
18 

9.76057 

0.15  182 

9.91  239 

12 

9.76075 

9.84  845 

27 
27 

0-15  155 

9.91  230 

48 

13 

9.76093 

9.84  872 

0.15  128 

9.91  221 

47 

I 

1.8 

1-7 

14 

9.76  1 11 

9.84  899 

^/ 
26 
27 
27 
27 
27 
26 
27 
27 

27 
26 

27 
27 

27 

26 

27 
27 

27 

26 

27 

0.15  lOI 

9.91  212 

9 
9 
9 
9 
9 
9 
9 

9 
s 

46 

2 

3.6 

3-4 

i.S 

9.76129 

17 
18 

18 

9.84  925 

0.15075 

9.91  203 

45 

3 

5-4 

8.5 

10.2 

i6 

9.76  146 
9.76  164 

9.84  952 
9.84  979 

0. 1 5  048 
0.15  02  [ 

9.91  194 
9.91  185 

44 
43 

4 

7-2 

9.0 

10.8 

i8 

9.76  182 

18 

9.85  006 

0.14994 

9.91  176 

42 

I 

9 

12.6 

11.9 
13-6 
15-3 

19 
20 

9.76  200 

18 

9-85  033 
9.85  059 

0.14967 

9.91  167 

41 
40 

14.4 
16.2 

9.76  218 

0.14  941 

9.91  158 

21 

9.76  230 

17 
18 

9.85  086 

0.14  914 

9.91  149 

39 

22 

9.76  253 

9.85113 

0.14887 

9.91  141 

9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 

38 

10   8 

8 

23 

24 

9.76271 
9.76  289 

18 
18 

9.85  140 
9.85  166 

0.14  860 
0.14  834 

9.91  132 
9.91  123 

37 
36 

I 
2 

1.0  0. 
2.0  I. 

9  0.8 
8  1.6 

25 

9.76  307 

17 

iS 

9-«5  193 

0.14  807' 

9.91  114 

35 

3 
4 

3.0  2. 
4.0  3. 

7  2.4 
6  3.2 

26 

9.76  324 

9.85  220 

0.14  780 

9.91  105 

34 

27 

9.76  342 

18 

9.85  247 

0.14753 

9.91  096 

33 

5.0  4.5  4.0  1 

28 

9.76  360 

18 

9-S5  273 

0.14727 

9.91  087 

32 

6 

6.0  5.4  4.8   1 

29 
30 

31 

9.76378 

17 

18 

18 

9.85  300 

0.14  700 

9.91  078 

31 
30 

29 

7 
8 

9 

7.0  6, 
8.0  7. 
9.0  8. 

3  5-6 

2  6.4 
I  7.2 

9.76  3c,5 

9-85  327 

0.14673 

2^1  069 

9-76413 

9-85  354 

0.14  646 

9.91  060 

32 

9.76431 
9.76  448 

17 

9.85  380 
9.85  407 

0.14620 

9-91051 

28 

33 

18 

0.14  593 

9 

27 

34 

9.76466 

18 

9-85  434 

27 
26 

0.14566 

9-91  033 

26 

3S 

9.76  484 

17 

18 

9.85  460 

0.14  540 

9.91  023 

9 
9 
9 

25 

10 

10 

36 

9.76501 

9.85  487 

27 

27 
26 

0.145J3 

9.91  014 

24 

27 

26 

37 

9.76519 

18 

9-85  5.H 

0.14486 

9.91  005 

23 

0 

,  1.4 

1-3 
3-9 
6.5 
9.1 
11.7 

3^ 

9-76537 

17 
18 

18 

9.85  540 

0.14460 

9.90  996 

22 

I 

39 
40 

41 

9-76  554 

9.85  567 
9.85  594 

27 
27 
26 

0.14433 

9-90  987 

9 
9 
9 

21 
20 

IQ 

2 

3 
4 

4.0 
6.8 
9-4 

12.2 

9.76572 

0.14406 

9.90  978 

9.76  590 

9.85  620 

0.14380 

9.90  969 

42 

9.76  607 

18 

9.85  647 

27 
27 
26 

27 
27 
26 

27 

0.14353 

9.90  960 

9 
9 
9 
9 
9 
9 

9 
10 

9 
9 
9 
9 

18 

5 
6 

7 
8 

14.8 

14-3 

43 

9.76625 

17 

18 

9.85  674 

0.14  326 

9.90951 

17 

17.6 

16.9 

44 

9.76  642 

9.85  700 

0.14300 

9-90  942 

16 

20.2 

19.5 

4S 

9.76  660 

17 

18 

9.85  727 

0.14273 

9-90  933 

15 

9 
10 

23.0 

22.1 

46 

9.76677 

9.85  754 

0.14  246 

9.90  924 

14 

25.6 

24.7 

47 

9.76695 

17 

18 

17 

18 

9.85  780 

0.14  220 

9.90915 

13 

•- 

48 

9.76712 

9.85  807 

0.14193 

9.90  906 

12 

± 

9 
26 

1.4 

49 
50 

51 

9.76  730 

9.85  834 

11 
27 
26 
27 

0.14  166 

9.90  896 

II 
10 

9 

9.76747 

9.85  860 
9.85  887 

0.14  140 

9.90887 

0 

7ii 

1-5 

9.76  765 

0.14  113 

9.90  878 

S2 

9.76  782 

9-85913 

0,14087 

9.90  869 

8 

4-5 

4-3 

S3 

9.76  800 

9.85  940 

0.14  060 

9.90  860 

7 

3 
4 

7-5 

7.2 

54 

9.76817 

^7 

18 

9.85  967 

26 

0.14033 

9.90851 

9 
9 

6 

10.5 

lO.I 

55 

9-76835 

9-85  993 

0.14007 

9.90  842 

s 

5 
6 

7 

13-5 

13.0 

56 

57 

9.76852 
9.76870 

17 
18 

17 

9.86  020 
9.86  046 

27 
26 
27 

26 

0.13  980 
0.13954 

9.90832 
9-90  823 

9 
9 

4 
3 

16.5 

19-5 
22.5 

25-5 

159 
18.8 
21.7 
24.6 

S8 

9.76887 

9.86073 

0.13927 

9.90  814 

2 

8 

59 
60 

9-76  904 

;^ 

9.86  100 

0.13900 

9.90  805 

9 
9 

0 

9 

9.76922 

9.86  126 

0.13874 

9.90  796 

1  L.Cos. 

1  d.  1  L.Cot.  |c.  d.|  L.Tan. 

L.  Sin. 

d.  1  ' 

1      P.P 

• 

.^ZL^ 


62 


/ 

1    L.  Sin.    !  d. 

1   L.  Tan.  |c.  d.j    L.  Cot. 

1    L.  Cos. 

Jd,|    - 

P.P. 

0 

I 

9.76922 

1 

-    17 
18 

9.86126 

27 
26 

0.13874 

9-90  796 

9 
10 

60 

S9 

976939 

9.86153 

0.13  847 

9.90  787 

2 

9-76957 

17 

■17 

18 

9.86  179 

0.13  821 

9-90  777 

58 

3 

9.76974 

9.86  206 

27 
26 

26 

0.13  794 

9.90  768 

9 

57 

4 

9.76991 

9.86  232 

0.13768 

9-90  759 

9 

S6 

27       26 

5 

9.77  009 

17 

17 
18 

9.86259 

0.13  741 

9.90  750 

9 

55 

2.7       2.6 

5.4      5-2 
8.1       7.8 

6 

7 

9.77026 
977043 

9.86  285 
9.86312 

27 
26 

0.13  715 

0.13688 

9.90  741 
9.90  731 

9 
10 

54 

53 

2 
3 

8 

9.77061 

17 
17 
17 

18 

9.86338 

27 
27 

26 
26 

0.13662 

9.90  722 

9 

S2 

4 

10.8     10.4 

9 
10 

II 

9.77078 

9.86  365 

0.13635 

9.90713 

9 

9 

10 

51 

50 

4Q 

7 

13-5     13.0 
16.2     15.6 
18.9     18.2 

977095 

9.86  392 

0. 1 3  608 

9.90  704 

9.77112 

9.86418 

0.13582 

9.90  694 

12 

977  130 

17 
^7 
17 

18 

9.86445 

0-13555 

9-90  685 

9 

48 

8 

21.6    20.8 

13 

977  147 

9.86471 

27 
26 

0.13529 

9.90  676 

9 

47 

9 

24.3     23.4 

14 

977  164 

9.86  498 

0.13  502 

9.90  667 

9 

46 

'5 

977  181 

9.86  524 

27 

26 

26 

27 
26 

27 
26 

27 
26 

0.13476 

9.90057 

45 

i6 

977  199 

17 

17 
17 
18 

17 
17 
17 
17 
17 
17 
17 
iS 

9.86551 

0.13449 

9.90  648 

9 

44 

17 

977216 

9.86577 

0.13423 

9-90  639 

9 

4^ 

18      17      16 

18 

977  233 

9.86  603 

0-13397 

9.90  630 

9 

42 

I 

1.8     1.7     1.6 
3-6     ^A    3-2 
5.4     5-1     4.8 
7.2     6.8     6.4 
9.0    8.5     8.0 

19 
20 

21 
22 

977250 

9.86  630 

0.13370 

9.90  620 

9 

9 

10 

41 
40 

39 
S8 

2 
3 
4 
5 

977  268 

9.86  656 

0.13  344 

9.90  61 1 

977  285 
977  302 

9.86  683 
9.86  709 

0-13  317 
0.13  291 

9.90  602 
9-90  592 

23 

977319 

9.86  736 

0.13264 

9-90583 

y 

37 

6     10.8  10.2     9.6 

24 

977336 

9.86  762 

0.13238 

9-90574 

9 

S6 

7     12.6  11.9  11,2 

2,S 

977  353 

9.86  789 

27 
26 

0.13  211 

9-90  565 

9 

35 

0     I 

4.4   13.0   I2.» 
6.2  15.3  14.4 

26 

977  370 

9.86815 

0.13  185 

9-90  555 

34 

9     i 

27 

9  77  3B7 

9.86  842 

27 
26 
26 
27 
26 
27 
26 

0.13  158 

9.90  546 

9 

33 

28 

977  405 

17 
17 
17 
17 
17 
17 
■17 
17 
17 
17 
17 
17 
17 
17 
17 
17 
17 
17 
17 
16 

9.86  868 

0.13  132 

9-90537 

9 

32 

29 

30 

31 

9.77422 

9.86  894 

0.13  106 

9.90527 

9 
9 

31 
30 

29 

10      9 

977  439 

9.86921 

0.13079 

9.90518 

977456 

9.86  947 

0.13053 

9.90  509 

I 

1 .0    0.9 

32 

977  473 

9.86  974 

0.13026 

9-90499 

9 
10 

9 
9 

28 

!     2.0     1.8 

33 
34 

977490 
977  597 

9.87  000 
9.87027 

27 
26 

0.13000 
0.12973 

9.90490 
9.90  480 

27 
26 

4 

.  3.0  2.7 

r       40        3.6 

35 

977  524 

9-87053 

26 

0.12947 

9.90471 

25 

5 

5.0    4-5 

3b 

977541 

9.87079 

0.12  921 

9.90  462 

24 

fc 

6.0    5.4 

37 

977558 

9.87  106 

27 
26 
26 
27 
26 
27 
26 
26 

0.12894 

9-90  452 

23 

7.0    6.3 

3« 

977  575 

9.87  132 

0.12868 

9-90  443 

9 

22 

8.0    7.2 

39 
40 

41 

977592 

9.87  158 

0.12842 

990  434 

9 
10 

9 

21 
20 

19 

9 

9.0    8.1 

977  609 

9.87  185 

0.12815 

9-90  424 

9.77  626 

9.87  21 1 

0.12789 

9.90415 

42 
43 

977  643 
9.77  660 

9.87  238 
9.87  264 

0.12  762 
0.12  736 

9.90  405 
9.90  396 

9 

18 
17 

44 

977  677 

9.87  290 

0.12  710 

9.90  386 

16 

45 

977  694 

987317 

27 
oft 

0.12683 

9-90377 

9 

15 

46 

977  7" 

9.87  343 

26 

0.12657 

9.90  368 

9 

lO' 

14 

9         9 

47 

9.77728 

9.87  369 

27 

26 

0.12  631 

9-90358 

13 

9.7          OR 

48 

977  744 

17 
17 
17 
17 
17 
17 
t6 

9.87  396 

0.12  604 

9-90  349 

9 

12 

49 
50 

51 

52 

53 
54 

9.77761 

9.87422 

26 
27 
26 
26 

27 

26 

0.12578 

990  339 

9 
10 

9 
10 

9 
10 

II 

10 

9 
8 

7 
6 

I 
2 

3 

4 

I 

1-5       1-4 

4-5       4-3 

7:5       7-2 

10.5     10. 1 

13-5     13.0 
16.5     15-9 
19.5     18.8 

9.77778 

9.87448 

0.12552 

9.90  330 

977  795 
9.77812 
977  829 
977  846 

9-87475 
9.87  501 
9.87  527 

9.87  554 

0.12525 
0.12499 
0.12473 
0.12446 

9.90  320 
9.90311 
9-90  301 
9.90  292 

55 

9.77  862 

17 

17 

17 

9.87  580 

26 

•0,12420 

9.90  282 

9 

1  r\ 

5 

f. 

22.5     21.7 
25.5     24.6  ' 

5^ 

977879., 

9.87  606 

27 

26 

0.12394 

9.90  273, 

4 

57 

977  896' 

9-87  633 

0.12  367 

9.90  263 

9 

3 

9      -  - 

5« 

977913 

9-87  659 

26 

0.12  341 

9.90  254 

2 

59 
60 

977  930 

9-87  685 

26  . 

0.12  315 

9.90  244 

9 

I 
0 

9.77946 

9.87  711 

0.12  289 

9-90  235 

1   L.  Cos.    !  d.  1 

L.  Cot.    |c.  d.l    L.  Tan.  | 

L.  Sin.     1 

d. 

/ 

P.P. 

63' 


7J^ 

mr 

<^3 

/ 

L.  Sin. 

d. 

L.Tan. 

c.  d. 

L.  Cot. 

L.  Cos. 

d. 

P.P.      1 

o 

_9:77  946_ 
9-77  9^>3 

n 

16 

17 
17 
16 

9.87  711 
9.87  738 

27 
26 

0.12  289 

9.90  235 

10 

9 
10 

60 

59 

0.12  262 

9.90  225 

2 

9.77  9i>o 

9.87  764 

26 

0.12  236 

9.90  216 

58 

3 

9.77  997 

9.87  790 

27 
26 

0.12210 

9.90  206 

9 
10 

57 

4 

^9.78013 

9.87817 

0.12  183 

9.90  197 

56 

27   26 

S 

9.78  030 

9.87  843 

26 

0.12157 

9.90187 

9 
10 

55 

I 

2.7   2.6 

6 

9.78047 

9.87  869 

26 

0.12  131 

9.90178 

54 

2 

5--^   5-2 

7 

9.78063 

9.87  895 

27 
26 
26 
26 

27 
?6 

0.12  105 

9.90  168 

9 
10 

53 

3 

8.1   7.8 
10.8  10.4 

8 

9.78  080 

17 

9.87  922 

0.12078 

9.90159 

52 

4 

9 
10 

II 

9.78  097 

17 
17 

t6 

9.87  948 

0.12052 

9.90  149 

lO 

9 
10 

51 
50 

49 

5 
6 

7 
8 

13.5  13.0 
16.2  15.6 
18.9  18.2 

21.6  20.8 

978  "3 

9.87  974 

0.12026 

9.90  139 

9.78  130 

9.88000 

0.12000 

9.90  130 

12 

9.78  147 

9.88027 

0.1 1  973 

9.90  120 

9 

48 

9 

24.3  23.4 

13 

9.78163 

9.88053 

26 
26 

0.1 1  947 

9.90  III 

10 
10 

47 

H 

9.78  180 

'7 

17 
16 

9.88  079 

0.1 1  921 

9.90  10 1 

46 

IS 

9.78  197 

9.88  105 

26 

0.11895 

9.90091 

9 

45 

i6 

9.78213 

9.88  131 

0. II  869 

9.90  082 

44 

17 

9.78  230 

16 
17 
17 
16 

9.88158 

27 
26 

0.1 1  842 

9.90072 

9 

43 

17   16 

18 

9.78246 

9.88  184 

'>6 

o.ii  816 

9.90063 

42 

I 

1.7   1.6 

19 
20 

21 

9.78  263 

9.88210 

26 
26 
27 
26 
26 
26 

0.1 1  790 

9.90053 

10 
9 

41 
40 

39 

2 
3 
4 

3.4  3.2 
5.1   4.8 
6.8   6.4 

8.5  8.0 

10.2  9.6 
11.9  11.2 
13.6  12.8 

15.3  14.4 

9.78  280 

9.88  236 

o.ii  764 

9.90  043 

9.78  296 

9.88  262 

O.II  738 

9.90  034 

22 
23 

24 

9-78313 
9.78  329 

9.78  346 

17 
16 

17 
16 

9.88  289 
9.88315 
9.88  341 

O.II  711 
O.II  685 
O.II  659 

9.90  024 
9.90014 
9.90  005 

10 

9 

38 
37 
36 

I 

2S 

9.7^362 

17 
16 

9.88  367 

26 

O.II  633 

9.89  995 

10 
9 

3S 

9 

26 

9-78  379 

9-88  393 

27 
26 
26 
26 
26 
96 

O.II  607 

9.89  985 

34 

27 

9.78  395 

9.88  420 

O.II  580 

9.89  976 

33 

28 

9.78412 

17 

16 

17 
16 

17 

16 

16 

9.88  446 

O.II  554 

9.89  966 

32 

29 

30 

31 

9.78428 

9.88472 

O.II  528 

9.89  956 

9 
10 
10 

31 
30 

29 

10   9 

9.78445 

9.88  498 

O.II  502 

9.89  947 

9.78461 

9.88  524 

O.II  476 

9.89  937 

I 

i.o  0.9 

32 

9.78478 

9^8550 

27 
26 
26 
26 
26 
26 
26 
26 
27 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 

O.II  450 

9.89  927 

28 

2 

2.0  1.8 

33 

9.78  494 

9.88577 

O.II  423 

9.89  918 

9 
10 

27 

3 

3.0  2.7 

34 

9.78510 

9.88  603 

O.II  397 

9.89  908 

26 

4 

4.0  3.6 

3S 

9.78  527 

17 
16 

9.88  629 

O.II  371 

9.89  898 

25 

5.0  4.5 

36 

9.78  543  ■ 

9.88  655 

O.II  345 

9.89  888 

24 

6 

6.0  5.4 

37 

9.78  560 

17 
16 
16 
17 
16 

9.88  681 

O.II  319 

9.89  879 

9 

23 

I 

7.0  6.3 
8.0  7.2 

38 

9.78576 

9.88  707 

O.II  293 

9.89  869 

22 

39 
40 

41 

9.78592 

9.88  733 

O.II  267 

9.89859 

10 
9 

21 
20 

19 

9 

9.0  8.1 

9.78  609 

9.88  759 

O.II  241 

9.89849 

9.78  625 

9.88  786 

O.II  214 

9.89  840 

42 
43 

9.78642 
9.78658 

17 
16 

16 

9.88812 
9.88  838 

O.II  188 

O.II  162 

Q.8q  8W 

10 

18 
17 

9.89  820 

44 

9.78  674 

9.88  864 

O.II  136 

9.89  810 

16 

4S 

9.78691 

17 
16 

16 

16 

9.88  890 

O.II  1 10 

9.89  801 

9 

15 

10   10 

46 

9.78  707 

9.88916 

O.II  084 

9.89  791 

14 

47 

9.78  723 

9.88  942 

O.II  058 

9.89  781 

13 

Zl        Zb 

48 

978  739 

9.88  968 

O.II  032 

9.89771 

12 

1.4   1.3 

49 
50 

51 

9.78756 

17 
16 

16 

9.88  994 

O.II  006 

9.89  761 

9 
10 

10 

9 

2 

3 
4 

4.0   3-9 
6.8   6.5 
9.4   9.1 

9.78  772 

9.89  020 

0.10980 

9.89  752 

9.78  788 

9.89  046 

0.10954 

9.89  742 

52 

9.78805 

16 
16 
16 
16 

17 
16 
16 
16 

9.89  073 

^7 
26 

26 

26 

0.10927 

9.89  732 

8 

1 

12.2  II. 7 

53 
S4 

9.78821 
9.78837 

9.89  099 
9.89125 

0.I090I 
0.10875 

9.89  722 
9.89712 

10 

7 
6 

14.8  14.3 
17.6  16.9 
20.2  19.5 
23.0  22.1 
25.6  24.7 

SS 

9.78853 

9.89  151 

26 
26 
26 
26 
26 

0.10849 

9.89  702 

S 

8 

56 
S7 

9.78  869 
9.78  886 

9.89  177 
9.89  203 

0.10823 

0.10797 

9.89  693 
9.89  683 

9 
10 

4 
3 

9 
10 

S« 

9.78  902 

9.89  229 

o.io  771 

9.89  673 

2 

59 
60 

9.78918 

9.89255 

0.10745 

9.89  663 

10 

I 
0 

978  934 

9.89  281 

O.IO  719 

9.89  653 

L.  Cos. 

d. 

L.  Cot. 

c.  d. 

L.  Tan. 

L.Sin. 

d. 

/ 

P.P. 

.^9/ 


64 


L.  Sin. 


L.  Tan.  c.  d 


L.  Cot. 


L.  Cos. 


P.  P. 


O 

I 

2 

3 

4 

5 
6 

7 
8 

9 

10 

II 

12 
13 
14 
15 
16 

}1 
18 

19 

20 

21 

22 

23 

24 

25 
26 

27 
28 
29 

30 

31 
32 
2,Z 
34 
35 
36 

37 
38 
39 
40 

41 

42 

43 
44 
45 
46 

47 
48 

49 

50 

51 

52 
53 
54 
55 
56 

57 
58 
59 
60 


78934 


78950 
78967 
78983 

78  999 
79015 
79031 
79047 
79063 

79  079 


79095 


79  in 
79  128 
79  M4 
79  160 
79176 
79  192 
79  208 
79224 
79240 


79  256 


79272 
79288 
79304 
79319 
79  335 
79351 
79367 
79383 
79  399 


9- 
9- 
9- 

9: 

9-79415 
9-79  431 
9-79  447 
9-79  463 
9.79  478 

9-79  494 
9.79510 

79526 
79542 
79558 


79  573 


79589 
79605 
79  621 

79636 
79652 
79668 

79684 
79699 
79715 


9- 

9: 

9i79  731 

9- 

9- 

9- 

9- 


-79  746 
-79  762 
-79  778 


79  793 
79809 
79825 

79840 
79856 
79872 


79887 


9.89  281 


9.89  307 
9-89  2>Z3 
9-89  359 

9-89  385 
9.89  41 1 

9-89437 

9-89  463 
9.89  489 

989515 


9.89  541 


9-89  567 
9-89  593 
9.89  619 

9-89  645 
9.89  671 

9-89  697 
9.89  723 
9.89  749 
9-89  775 
9.89  801 


9.89827 

9-89  853 
9.89  879 

9.89  905 
9.89931 
9-89957 

9-89  983 

9.90  009 
9-90035 


9.90  06.1 
9.90  086 
9.90  112 
9-90  138 
9.90  164 
9.90  190 
9.90216 
9.90  242 
9.90  268 
9.90  294 


9,90  320 


9.90  346 
9.90371 
9-90  397 
9.90423 
9-90  449 
9.90475 

9.90  501 
9.90527 
9-90553 


9-90  578 
9.90  604 
9-90  630 
9.90  656 
9.90  682 
9.90  708 
990  734 

9-90  759 
9.90  785 
9.90  811 


9.90  837 


o.io  719 


0.10  693 
o.io  667 
O.IO  641 

O.I06I5 
O.IO  589 

0.10563 
0.10537 

O.IO  511 
0.10485 
0.10459 


9-89  653 
9-89  643 
989  633 

9.89  624 
9.89  614 
9.89  604 
9.89  594 

9.89  584 
9-89  574 
9.89  564 


0.10433 
0.10407 
O.IO  381 

0-10355 

O.IO  329 

0.10303 

O.IO  277 
O.IO  251 
O.IO  225 


O.IO  199 


O.IO  173 
O.IO  147 

O.IO  121 

0.10095 
0.10069 

0.10043 

O.IOOI7 
0.09  991 
0.09  965 
0.09  939 


9  ■895  51 
9-89  544 
9-89  534 
9-89  524 
9.89514 
9.89  504 
9.89  495 
9.89  485 
9-89  475 
9-89465 
9-89455 
9-89  445 
989435 
9.89425 

9.89415 
9.89  405 
9-89  395 
9-89  385 
989  375 
9-89  364 


9-89354 


0.09914 
0.09  888 
0.09  862 
0.09  836 
0.09  810 
0.09  784 
0.09  758 
0.09  732 
0.09  706 


9-89  344 
9-89  334 
9.89  324 

9.89314 
9.89  304 
9.89  294 
9.89  284 
9.89  274 
9.89  264 


0.09  680 


9.89  254 


0.09  654 
0.09  629 
0.09  603 
0.09  577 
0.09551 
0.09  525 

0.09  499 
0.09  473 
0.09  447 


9.89  244 
9  89  233 
9.89  223 
9.89213 
9.89  203 
9-89  193 
9.89  183 
9.89  173 
9.89  162 


0.09  42i 


0.09  396 
0.09  370 
0.09  344 
0.09  318 
0.09  292 
0.09  266 
0.09  241 
0.09  215 
0.09  189 


9.89  152 
9.89  142 
9.89  132 
9.89  122 
9.89  112 
9.89  lOI 
9.89091 
9.89081 
9.89071 
9.89  060 


0.09  163 


9.89050 


60 

59 
58 
57 
56 
55 
54 

53 
52 
51 
50 

49 
48 
47 
46 
45 
44 

43 
42 
41 
40 

39 
38 
31 
36 

35 
34 

32 
31 
30 

29 
28 
27 
26 

25 

24 

23 
22 
21 
20 

19 
18 

17 
16 

15 
14 

13 
12 
II 

10 

9 

8 

7 
6 
5 
4 

3 
2 
I 

O 


26 

2.6 
5-2 
7-8 
10.4 
13.0 
15.6 
18.2 
20.8 
23.4 


25 

2.5 

5-0 

7-5 

1 0.0 

12.5 

15.0 

17-5 
20.0 

22.:; 


17  16  15 


1-7 

3-4 

5-1 

6.8 

8.5 

10.2 

11.9 

13-6 

15-3 


1.6 

3-2 

4-8 

6.4 

8.0 

9-6 

II. 2 

12.8 

14.4 


1-5 

3-0 

4-5 
6.0 

7-5 

9-0 

10.5 

12.0 

13-5 


11  10  9 


I.I 
2.2 
?>■?, 
4-4 

5-5 
6.6 

7-7 


i.o 
2.0 
3-0 
4.0 

5-0 
6.0 

7-0 
8.0 

9-9  9-0 


0.9 
1.8 

2.7 
3-6 
4.5 
5-4 
6.3 
7.2 
8.1 


10 
26 

1-3 

3-9 

6-5 

9.1 

11.7 

14-3 


10 
25 

1.2 

3-8 
^6.2 

8.8 
II. 2 


1.4 

4-3 
7.2 
0.1 


16.9  16.2 
19-5 


13-8  15-9 


18.8 


22.1 


18.8  21.7 
21.2  24.6 


24.7  23.8  — 


L.  Cos. 


L.  Cot.  c.  d.  L.  Tan. 


L.  Sin.  I  d. 


P.  P. 


Kl^ 


39' 


65 


t 

L.  Sin. 

d. 

L.  Tan. 

c.  d. 

L.  Cot. 

L.  Cos. 

d. 

P.P. 

0 

I 

9.79  887 

16 

15 
16 

9.90  837 

26 

26 

25 
26 
26 
26 
26 

25 
26 
26 
26 
26 

0.09  163 

9.89  050 

10 
10 

60 

59 

9.79  903 

9.90  863 

0.09  137 

9.89  040 

2 

9.79918 

9.90  889 

0.09  1 1 1 

9.89  030 

58 

3 

9-79  934 

16 

11 

9.90914 

0.09  086 

9.89  020 

57 

26   26 

4 

9-79  950 

9.90  940 

0.09  060 

9.89  009 

56 

5 

979965 

9.90  966 

0.09  034 

9.88999 

ir» 

55 

I 

2.0   2,5 

6 

9.79  981 

9.90  992 

0.09  008 

9.88  989 

54 

2 

5.2   5.0 
7.8   7.5 
10.4  1 0.0 
13.0  12.5 
15.6  15.0 
18.2  17.5 
20.8  20.0 

7 

9.79  996 

^5 
16 

\i 

15 
16 

9.91  018 

0.08  982 

9.88  978 

10 

II 

S3 

3 

8 

9.80012 

9.91  043 

0.08957 

9.88  968 

52 

4 

7 
8 

9 

io 

II 

9.80027 

9.91  069 

0.08931 

9.88  958 

51 
50 

49 

9.80  04.? 

9.91  095 

0.08  905 

9.88  948 

9.80  058 

9.91  121 

0.08  879 

9.88937 

12 

9.80  074 

15 

t6 

9.91  147 

25 
96 

0.08  853 

9.88927 

ir> 

48 

9 

23.4  22.1: 

13 

9.80  089 

9.91  172 

0.08  828 

9.88917 

10 

47 

14 

9.80  105 

\l 

9.91  198 

26 

0.08  802 

9.88  906 

46 

^5 

9.80  1 20 

9.91  224 

26 

0.08  776 

9.88  896 

45 

16 

9.80  136 

15 

15 
16 

9.91  250 

26 

0.08  750 

9.88  886 

10 

44 

16   15 

17 

9.80  151 

9.91  276 

25 
26 

0.08  724 

9.88875 

43 

I 

1.6   1.5 

18 

9.80  ibb 

9.91  301 

0.08  699 

9.88  865 

42 

2 

3-2   3-0 

19 
20 

21 

9.80  182 

15 
16 

9.91  327 

26 
26 

'4 

0.08  673 

9.88  855 

II 

10 

41 
40 

^9 

3 

4 

4.8   4.5 
6.4   6.0 

8.0   7.5 

9.80  197 

9.91  353 

0.08  647 

9.88  844 

9.80213 

9-91  379 

0.08  621 

9.88  834 

22 

9.80  228 

15 

16 

9.91  404 

0.08  596 

9.88  824 

11 

10 

38 

6 

9.6   9.0 

23 

24 

9.80  244 
9.80  259 

15 

9.91  430 
9.91  456 

26 
26 

25 
26 

0.08  570 
0.08  544 

9.88813 
9.88  803 

37 
36 

7 
8 

9 

1 1.2  10.5 
12.8  12.0 
14.4  13-5 

25 

9.80  274 

15 

9.91  482 

0.08518 

9.88  793 

35 

2b 

9.80  290 

9.91  507 

0.08  493 

9.88  782 

34 

27 

9.80  305 

9.91  533 

26 

0.08  467 

9.88  772 

33 

28 

9.80  320 

9.91  559 

26 

0.08441 

9.88  761 

10 
10 
II 
10 

32 

11   in 

29 

30 

31 

9.80336 

15 

15 
16 

9-91  585 

25 
26 
26 

0.08  4 1 5 

9.88751 

31 
30 

29 

I 

2 
3 
4 

5 
6 

1.1  I.O 

2.2  2.0 

3-3    3.0 

4.4  4.0 

5-5  5-0 
6.6  6.0 

9.80351 

9.91  610 

0.08  390 

9.88  741 

9.80  36b 

9.91  636 

0.08  364 

9.88  730 

32 

9.80  382 

15 
15 
16 

15 
15 

9.91  662 

26 

0.08  338 

9.88  720 

28 

33 
34 

9.80  397 
9.80412 

9.91  688 
9-91  713 

25 
26 
26 
26 

0.08  3 1 2 
0.08  287 

9.88  709 
9.88  699 

10 

27 
26 

35 

9.80  428 

9-91  739 

0.08  261 

9.88  688 

25 

7 

7.7  7.0 

3^ 

9.80  443 

9.91  765 

0.08  235 

9.88  678 

24 

8 

8.8  8.0 

37 

9.80458 

9.91  791 

0.08  209 

9.88  668 

23 

9 

9.9  9.0 

3« 

9.80  473 

15 
16 

15 

15 

9.91  816 

11 
26 

25 
26 
26 
26 

0.08  184 

9.88  657 

^' 

22 

39 
40 

41 

9.80  489 

9.91  842 

0.08  158 

9.88  647 

II 

21 
20 

19 

9.80  504 

9.91  868 

0.08  132 

9.88  636 

9.80519 

9.91  893 

0.08  107 

9.88  626 

42 

9-8o  534 

9.91  919 

0.08081 

9.88615 

^^ 

18 

43 

9.80  550 

15 

9-9 1  945 

0.08  055 

9.88  605 

M 

17 

44 

9.80565 

9.91  971 

0.08  029 

9.88  594 

16 

45 

9.80  580 

^5 
15 
15 

15 
16 

9.91  996 

25 
26 

26 

0.08  004 

9.88  584 

lu 

15 

11   11 

4b 

9-8o  595 

9.92022 

0.07  978 

9-88573 

^' 

H 

26   25 

47 

9.80  610 

9.92  048 

0.07  952 

9.88563 

13 

0 

1.2   I.I 

48 

9.80  625 

9.92073 

0.07  927 

9.88552 

" 

12 

I 

3.5   3.4 

49 
50 

51 

9.80  641 

15 
15 
15 
15 
15 
15 
15 
16 

15 
15 
15 

9.92  099 

26 

25 
26 
26 

0.07  901 

9.88  542 

10 

II 
10 

9 

2 
3 
4 

S.s      8!o 
10.6  10.2 

9.80  656 

9.92  125 

0.07  875 

9.88531 

9.80671 

9.92  150 

0.07  850 

9.88521 

52 

9.80  686 

9.92176 

0.07  824 

9.88510 

8 

13.0  12.5 

53 

9.80  701 

9.92  202 

0.07  798 

9.88499 

7 

I 

15.4  14.8 

54 

9.80716 

9.92227 

25 
26 
26 

0.07  773 

9.88  489 

6 

17.7  17.0 

55 

9.80  731 

9-92  253 

0.07  747 

9.88478 

5 

9 

22.5  21.6 
24.8  23.9 

5^ 

9.80  746 

9.92  279 

0.07  721 

9.88  468 

4 

10 

57 

9.80  762 

9.92  304 

25 
76 

0.07  696 

9.88457 

10 

3 

II 

5^ 

9.80  777 

9.92  330 

26 
25 

0.07  670 

9.88447 

2 

59 
60 

9.80  792 

9.92  356 

0.07  644 

9.88436 

II 

0 

9.80  807 

9.92381 

0.07  619 

9.88  425 

L.  Cos. 

d. 

L.  Cot. 

c.  d. 

L.  Tan. 

L.  Sin. 

d. 

/ 

P.P. 

66 


'    L.  Sin.  1 

d.   L.  Tan.  |c.  d.|  L.  Cot.  | 

L.  Cos.  1 

d.|    1      P.P.      Il 

O 

I 

9.80  807 
9.80822 

15 

15 
15 
15 
15 
15 
15 

9.92  381 

26 
26 

25 
26 
26 

0.07  619 

9.88425 

10 

60 

S9 

9.92  407 

0.07  593 

9.88415 

2 

9.80837 

9.92  433 

0.07  567 

9.88  404 

ir\ 

58 

3 

9.80852 

9.92458 

0.07  542 

9.88394 

57 

26 

25 

4 

9.80  867 

9.92  484 

0.07  516 

9.88383 

J  J 

56 

I 

2.6 

2.5 

5 

9.80  882 

9.92510 

25 
26 

0.07  490 

9.88  372 

f  r> 

55 

2 

S.2 

5.0 

6 

9.80897 

9-92  535 

0.07  465 

9.88  362 

54 

3 

7.8 

7-5 

7 

9.80912 

9.92561 

26 

11 

25 
26 

0.07  439 

9.88351 

S3 

4 

10.4 

lO.O 

8 

9.80  927 

^5 

9.92  587 

0.07413 

9.88  340 

52 

^^ 

I3.U 

12.5 

9 
10 

9.80  942 

15 
15 
15 
15 
15 
15 
15 
15 

9.92  612 

0.07  388 

9.88330 

51 
50 

6 

7 
8 

15.6 
18.2 

20  8 

15.0 

17.5 

9.80957 

9.92638 

0.07  362 

9.88319 

II 

9.80972 

9.92  663 

0.07  337 

9.88308  1 

10 

49 

9 

234 

22.5 

12 

9.80  987 

9.92  689 

26 

25 
-76 

0.07  311 

9.88  298 

48 

13 

9.81  002 

9.92715 

0.07  285 

9.88  287 

47 

H 

9.81  017 

9.92  740 

0.07  260 

9.88276 

ir» 

46 

16 

9.81  032 
9.81  047 

9.92  766 
9.92  792 

26 

0.07  234 
0.07  208 

9.88  266 
9.88  255 

II 

45 

44 

15 

14 

17 

9.81  061 

14 
15 
15 
15 
15 
15 
15 

9.92817 

25 
26 

0.07 183 

9.88  244 

10 

43 

2 

3.0 

6.0 

7.5 
9.0 

1.4 
2.8 

i8 
19 
20 

21 

9.81  076 
9.81  091 

9.92  843 
9.92  868 

11 
26 

11 

0.07 157 
0.07 132 

9.88  234 
9.88  223 

II 

42 
41 
40 

39 

3 
4 

1 

4.2 
.5.6 
7.0 
8.4 

9.81  106 

9.92  S94 

0.07 106 

9.88212 

9.81  121 

9.92  920 

0.07  080 

9.88  201 

22 

9.81  136 

9-92  945 

0.07  055 

9.88  191 

I  I 

38 

7 

10.5 

9.8 

23 

9.81  151 

9.92971 

0.07  029 

9.88  180 

37 

8 

12.0 

II. 2 

24 

9.81  166 

^5 
14 
15 
15 
15 
15 
H 
15 
15 
15 
»5 
14 
15 
^5 

9.92  996 

25 
26 
26 

25 
26 

11 

25 
26 
26 

0.07  004 

9.88  169 

36 

9 

13-5 

12.6 

2=; 

9.81  180 

9.93  022 

0.06  978 

9.88  158 

ir» 

35 

26 

9.81  195 

9.93  048 

0.06952 

9.88  148 

J  J 

34 

27 
28 

9.81  210 
9.81  225 

993073 
9.93  099 

0.06  927 
0.06  901 

9.88137 
9:SSi26 

II 

33 
32 

11 

10 

29 

30 

31 

9.81  240 

9.93124 

0.06  876 

9.88 115 

10 

31 
30 

29 

I 
2 
3 

I.I 

2.2 
3-3 

I.O 

2.0 
3.0 

9.81  254 

9-93150 

0.06  850 

9.88  105 

9.81  269 

9-93  175 

0.06  825 

9.88  094 

32 

9.81  284 

9.93  201 

0.06  799 

9.88  083 

II 

28 

4 

4.4 

4.0 

33 
34 

9.81  "299 
9.81  314 

9.93  227 
9.93  252 

25 
26 

0.06  773 
0.06  748 

9.88072 
9.88061 

II 

27 
26 

7 
8 

7-7 
8  8 

5.0 
6.0 
7.0 
80 

3S 

9.81  328 

9.93  278 

25 
26 

0.06  722 

9.88051 

I  I 

25 

36 

37 

9.81  343 
9.81  358 

9-93  303 
9-93  329 

0.06  697 
0.06671 

9.88  040 
9.88  029 

II 

24 
23 

9 

9-9 

9.0 

38 

9.81  372 

14 
15 
15- 
15 

14 

9-93  354 

11 
26 

25 
26 

0.06  646 

9.88018 

22 

39 
40 

41 

9.81  387 

9-93  380 

0.06  620 

9.88007 

10 

21 
20 

19 

9.81  402 

9.93  406 

0.06  594 

9.87  996 

9.81417 

9.93431 

0.06  569 

9.87985 

42 

9.81  431 

9.93457 

25 
26 

0.06  543 

9.87975 

18 

43 

9.81  446 

15 

15 

9.93  482 

0.06518 

9.87  964 

17 

44 

9.81  461 

9.93  508 

0.06  492 

9.87  953 

16 

11 

10  10 

45 

9.81  475 

14 
15 

9.93  533 

25 
26 

0.06  467 

9.87  942 

II 

15 

OR 

26  25 

46 

9.81  490 

9.93  559 

0.06  441 

9.87931 

I  I 

14 

n 

47 

9.81  505 

15 

9-93  584 

25 
26 
26 
25 
26 

'4 

0.06416 

9.87  920 

I  I 

13 

I 

1.2 

1.3  1.2 

3.9  3.8 

48 

9.81  519 

14 

9.93  610 

0.06  390 

9.87  909 

I  I 

12 

2 

3.5 

49 
50 

51 

9.81  534 

15 
15 
14 

9-93  636 

0.06  364 

9.87  898 

II 
10 
I  I 

II 

io 

9 

3 
4 

5 

5.9  0.5  0.2 

8.3  9.1  8.8 

10.6  II. 7  II. 2 

170  M-  I'j.S 

9.81  549 

9.93  661 

0.06  339 

9.87  887 

9.81  563 

9.93  687 

0.06  313 

9.87877 

52 

9.81  578 

15 
H 
15 
15 

9.93712 

0.06  288 

9.87  866 

II 

a 

6 

15.4  I 
17.7  I 

6.9  16.2 

QX    18.8 

53 

9.81  592 

9-93  738 

25 
96 

0  06  262 

9.87  855 

II 

7 

7 
8 

54 

9.81  607 

9.93  763 

0.06  237 

9.87  844 

I  I 

6 

20.1  22.1  21.2 

55 

9.81  622 

9.93  789 

25 
26 

O.Ob  211 

9.87  833 

I  I 

5 

9 

22.5  24.7  23.8 

56 

9.81  636 

;4  9.93814 

0.06  186 

9.87  822 

I  I 

4 

24.8 

—   — 

57 

9.81  651 

15 

9.93  840 

11 

25 

0.06  160 

9.87  811 

3 

58  1  9.81  665 

14 

9-93  865 

0.06  135 

9.87  800 

2 

59 

9.81  680 

15 

14 

9.93891 

0.06  109 

9.87  789 

II 

0 

60 

9.81  694 

9.93916 

0.06  084 

9.87  778 

1  L.  Cos. 

1  d.  1  L.Cot.  |c.  d.j  L.Tan.  |  L.  Sin.  |  d.  |  '  |      P.P.      I] 

/IQ° 


^\^ 


67 


L.  Sin. 


L.  Tan.  c.  d.  L.  Cot. 


L.  Cos. 


P.  P. 


2 
3 

4 
5 
6 

7 

8 

9 
10 

II 
12 
^3 
14 
15 
16 

17 

18 

19 
20 

21 
22 
23 
24 
25 
26 

27 
28 
29 

30 

31 

32 
33 
34 
35 
36 

37 
38 
39 
40 

41 

42 
43 
44 
45 
46 

47 
48 

49 

50 

51 
52 

53 
54 
55 
56 

57 
58 
59 
60 


9,81  694 
9.81  709 
9.81  723 
9.81  738 

9.81  752 
9.81  757 
9.81  781 
981  796 
9.81  810 
9.81  825 


9-8 i  839 


9.81  854 
9.8I  868 
9.81  882 
9.81  897 
9.81  911 
9.81  926 
9.81  940 
9-8i  955 
9.81  969 

9.81  983 


9.81  998 
9.82012 
9.82026 
9.82041 

9.82  055 
9.82  069 
9  82  084 
9.82  098 
9.82  112 


9.82  126 


9.82  141 
9.82  155 
9.82  169 
9.82  184 
9.82  198 
9.82  212 
9.82  226 
9.82  240 
9.82  255 


9.82  2b9 


9.82  283 
9.82  297 
9.82  31 1 
9.82  326 
9.82  340 
9.82  354 
9  82  368 
9.82382 
9.82 


9.82  410 


9.82  424 
9.82439 
9.82453 
9.82  467 
9.82  48_^ 
9,82  495 
9.82  509 
9.82  523 
9-82537 


9-82551 


9.93916 


9-93  942 
9-93  967 
9-93  993 
9.94018 

9-94  044 
9.94  069 

9-94  095 
9.94  120 
9.94  146 


9.94  171 


9-94  197 
9.94  222 
9.94  248 

9-94  273 
9-94  299 
9-94  324 
9-94  350 
9-94  375 
9.94  401 


9-94  426 
9.94452 
9-94  477 
9-94  503 
9.94528 
9-94  554 
9.94  579 
9.94  604 
9.94  630 
9-94  655 


9.94681 


9-94  706 
9-94  732 
9-94  757 

9-94  783 
9.94  808 
9.94  834 

9-94  859 
9.94  884 
9.94910 


9-94  935 


9.94  9b  I 

9.94  986 

9.95  012 

9-95  037 
9.95  062 
9.95  088 

9-95  "3 
9  95  139 
9-95164 


9.95  190 


995215 
9.95  240 
9.95  266 
9.95  291 
9-95317 
9-95  342 
9-95  368 
9-95  393 
9.95  418 


9-95  444 


0.06  084 


0.06  058 
0.06  033 
0.06  007 
0.05  982 
0.05  956, 
0.05931 
0.05  905 
0.05  880 
0.05  854 


9-87  778 

9.87  767 
9.87  756 

9-87  745 
9.87  734 
9.87  723 
9.87712 

9,87  701 
9.87  690 
9-87  679 


0.05  829 


9.87  668 


0.05  803 
0.05  778 
0.05  752 
0.05  727 
0.05  701 
0.05  676 
0.05  650 
0.05  625 
0.05  599 


9.87  657 
9.87  646 
9-87  635 
9,87  624 
9.87613 
9.87  601 
9.87  590 

9-87579 
9.87  568 


0.05  574 


9-87557 


0.05  548 
0.05  523 
0.05  497 
0.05  472 
0.05  446 
0.05  421 
0.05  396 
0.05  370 
0-05  345 


9.87  546 

9-87  535 
9.87  524 

9-87513 
9-87  501 
9.87  490 

9.87  479 
9.87  468 
9-87457 


005319 


9-87  446 


0.05  294 
0.05  268 
0.05  243 
0.05  217 
0,05  192 
0.05  166 
0.05  141 
0.05  116 
0.05  090 


9-87  434 
9-87  423 
9.87412 

9.87  401 
9.87  390 
9-87  378 
9.87  367 
9-87  356 
9-87  345 


0.05  065 


9-87  334 


0.05  039 
0.05  014 
0.04  988 
0.04  963 
0.04  938 
0.04  912 
0.04  887 
0.04  86 1 
0.04  836 


9.87322 
9.87  31 1 
9-87  300 
9.87  288 
9.87  277 
9.87  266 

9-87  255 
9-87  243 
9.87  232 


0.04  8ro 


9.87  221 


0.04  785 
0.04  760 
0.04  734 
0.04  709 
0.04  683 
0.04  658 
0.04  632 
0.04  607 
0.04  582 


9.87  209 
9.87  198 
9.87187 

9-87175 
9.87  164 

987  153 
9.87  141 
9-87  130 
9.87  119 


60 

59 
58 
57 
56 
55 
54 

53 
52 
51 
50 

49 
48 
47 
46 
45 
44 

43 
42 
41 
40 

39 
38 
37 
36 
35 
34 

33 
32 
31 
30 

29 
28 
27 
26 

25 
24 

23 
22 
21 

20 

19 
18 

17 
16 
15- 
14 

13 
12 
II 

10 

9 
8 

7 
6 

5 
4 

3 
2 

I 


26   25 


2.6 

2.S 

5-2 

50 

7.8 

7-5 

10.4 

1 0.0 

13.0 

12.S 

15.6 

15.0 

18.2 

17-5 

20.8 

20.0 

23-4 

22.5 

15 


14 


I 

1-5 

1.4 

2 

30 

2.8 

3 

4-5 

4.2 

4 

6.0 

S.6 

5 

7-5 

7.0 

6 

9.0 

8.4 

7 

10.5 

9.8 

8 

12.0 

II. 2 

9 

13-5 

12.6 

12  11 


1.2 

2.4 
3-6 
4-8 
6.0 

7.2 

8.4 

9.6 

10.8 


I.I 

2.2 

3-3 
4.4 

5-5 
6.6 

7-7 
8.8 

9-9 


12  12  11 
26  25   25 


i.o 
3-1 
5-2 
7-3 
9.4 
11.5 

»3-5 
16.2  15.6  17.0 
18.4  17.7  19.3 
20.6  19.8  21.6 

22.8  21.9  239 

24.9  24.0  — 


I.I 

3-2 

5-4 

7.6 

9.8 

11.9 

14.1 


3-4 

5-7 

8.0 

10.2 

'2-5 
14.8 


0.04  556 


9.87  107 


L.Cos.  I  d. 


L.  Cot.  |c.  d.|  L.  Tan. 


L.  Sin.  I  d. 


P.P. 


o8 


# 

L.  Sin. 

d. 

L.  Tan. 

c.  d. 

L.  Cot. 

L.  Cos. 

d. 

p.p       i[ 

o 

I 

9.82551 

H 
14 
14 

9-95  444 

25 
26 

0.04  556 

9.87  107 

II 

60 

59 

9.82  565 

9.95  469 

0.04531 

9.87  096 

2 

9.82  579 

9.95  495 

25 

0.04  505 

9.87085 

1 2 

58 

3 

9.82  593 

9.95  520 

0.04  480 

9.87073 

57 

26 

25 

4 

9.82  607 

14 

14 

9-95  545 

25 

?6 

0.04  455 

9.87  062 

12 

56 

2  6 

2-5 

5 

9.82621 

9-95571 

25 
26 

25 
25 
26 

25 

0.04  429 

9.87050 

T  T 

55 

6 

7 

9-82635 
9.S2  649 

14 

995  596 
9-95  622 

0.04  404 
0.04  378 

9.87  039 
9.87028 

II 

54 

53 

3 
4 

10.4 

5.0 

7-5 
1 0.0 

a 

9.82  663 

^4 
14 
14 
H 

9-95  647 

0.04  353 

9.87016 

I  I 

52 

5 
6 

7 
8 

I'?.0 

12.5 

15.0 

'7-5 
20.0 

9 

\o 

II 

9.82677 

9.95  672 

0.04  328 

9.87005 

12 
II 

51 
50 

49 

15.6 

18.2 
[  20.8 

9.82691 

9.95  698 

0.04  302 

9.86993 

9.82  705 

9-95  723 

0.04  277 

9.86982 

12 

9.82719 

^4 
H 

9-95  748 

0.04  252 

9.86970 

1 1 

48 

9 

I23.4 

22.5 

13 

9-82  733 

9-95  774 

25 

■2  ft 

0.04  226 

9-86  959 

47 

H 

9.82  747 

^4 
14 
14 

9-95  799 

0.04  201 

9-86947 

1 1 

46 

:i 

9.82761 
9.82  775 

9.95  825 
9-95  850 

25 

0.04  1 75 
0.04  150 

9.86936 
9.86924 

12 
II 

45 
44 

14 

13 

17 

9.82  788 

^3 

9-95  875 

25 
26 

0.04  125 

9.86913 

43 

' 

'A 

^•3 

i8 

9.82  802 

^4 

9.95  901 

0.04  099 

9.86  902 

12 

1 2 

42 

2 

2.6 

19 
20 

21 

9.82816 

14 
14 
14 

9-95  926 

11 

25 
25 
26 

0.04  074 

9.86  890 

41 
40 

39 

3 
4 

I 

4.2 

.  5-6 
7.0 
8.4 

3-9 
5-2 
6.5 
7.8 
9.1 

9.82  830 

9-95  952 

0.04  048 

9.86879 

9.82844 

9-95  977 

0.04  023 

9.86  867 

22 

9.82858 

H 

9.96002 

0.03  998 

9.86855 

38 

7 

9.8 

23 

9.82872 

H 

9.96028 

0.03  972 

9.86  844 

37 

8 

II. 2 

10.4 

24 

9.82  885 

13 

9.96053 

25 

0.03  947 

9.86832 

T  T 

36 

9 

12.6 

11.7 

2S 

9.82  899 

M 

9.96078 

25 
26 

0.03  922 

9.S6821 

35 

26 

9.82913 

H 

9.96  104 

0.03  896 

9.86  809 

34 

27 

9.82927 

14 

9.96129 

25 

0.03871 

9.86  798 

33 

12 

11 

28 

9.82941 

H 

9.96  155 

0.03  845 

9.86  786 

I  I 

32 

29 

30 

31 

9.82955 

H 
13 
H 

9.96  180 

25 
25 

26 

0.03  820 

9.86775 

12 

II 

31 
30 

29 

2 

3 

1.2 

2.4 
.3-6 
4-8 
6.0 
7.2 
8.4 

I.I 

2.2 
3-3 

9.82  968 

9,96  205 

0.03  795 

9-86  763 

9.82982 

9.96  231 

0.03  769 

9.86752 

32 

9.82  996 

H 

9.96  256 

25 

0.03  744 

9.86  740 

28 

4 

4-4 

33 
-  34 

9.83010 
9.83023 

13 

9.96  281 
9.96  307 

25 
26 

25 

0.03719 
0.03  693 

9.86  728 
9J6^7- 

II 

27 

_26. 

I 

7 
8 

5-5 
6.6 

1-1 

1t. 

9-83037 

14 

996332 

0.03  668 

9.86  705 

25 

9-6 
10.8 

8.8 

^83051 

14 

9.96357 

25 
26 

0.03  643 

9.86  694 

24 

9 

9-9 

37 

9.$3o65 

9-96  383 

0.03617 

9.86682 

\: 

23 

38 

9.83  078 

ii 

9.96  408 

25 

0.03  592 

9.86  670 

\2 

,  T 

22 

39 
40 

41 

9.83  092 

14 
14 
14 

9-96433 

26 

25 
26 

0.03  567 

9.86  659 

12 

21 

20 

19 

9.83  106 

9-96  459 

0.03  541 

9.86647 

9.83  120 

9-96  484 

0.03  5 1 6 

9.86635 

42 

9-83  ^33 

13 

9.96510 

0.03  490 

986624 

12 

18 

43 

9-83  H7 

14 

9-96  535 

25 

0.03  465 

9.86  612 

17 

12   11   11 

44 

9.83  161 

H 

9.96  560 

25 
26 

0.03  440 

9.86  600 

16 

26  26  25 

4S 

9.83174 

^3 

9.96  586 

0.03414 

9.86  589 

15 

o| 

46 

9.83  188 

14 

9.96  611 

25 

0.03  389 

9-86577 

14 

li 

I.I  I 

2  I.I 

47 

9.83  202 

M 

9.96  636 

25 
26 

25 
25 
26 

25 

25 
26 

0.03  364 

9.86565 

13 

2j 

3-2  '3  5  3-4 

48 

983215 

M 

9.96  662 

0.03  338 

986554 

12 

3 

5-4  5-9  5.7 
7.6  8.3  8.0 

9.8  10.6  10.2 

1.9  13.0  12.5 

4.1  15.4  14.8 

49 
50 

51 

9.83  229 

14 
13 
14 

9.96  687 

0.03313 

9-86  542 

12 
12 

II 
10 

9 

4; 

|i 

9.83  242 

9.96712 

0.03  288 

9-86  530 

9.83  256 

9.96  738 

0.03  262 

9.86518 

52 

9.83  270 

14 

9.96  763 

0.03  237 

9.86  507 

1 2 

8 

Zii 

6.2  17.7  17.0 

53 

9-83  283 

13 

9.96  788 

0.03  212 

9-86  495 

12 

T  f 

7 

8.4  20.1  19.3 

54 

9.83  297 

14 

9.96814 

0.03  186 

9.86  483 

6 

in^ 

0.6  22 

5  21.6 

55 

9.83310 

13 

9-96  839 

25 

0.03  161 

9.86472 

5 

l^ 

2.8  24 

8  23.9 

56 

9-83  324- 

H 

9.96  864 

25 
26 

0.03  136 

9.86460 

T  0 

4 

12^ 

4.9  - 

— 

57 

9-83  338 

14 

9.96  890 

0.03  no 

9.86448 

3 

5« 

9-83351 

13 

9.96915 

25 

0.03  085 

9.86436 

2 

59 
60 

9-83  365 

14 
13 

9-96  940 

11 

0.03  060 

9.86425 

12 

0 

9.83  378 

9.96966 

0.03  034 

9.86413 

L.  Cos. 

d. 

L.  Cot. 

c.  d.i  L.Tan. 

L.  Sin. 

d. 

/ 

P.  P 

i\y^o 


43° 

6S 

> 

/ 

L.  Sin.   d.  1 

L.Tan.  |c.  d.|  L.  Cot.  |  L.  Cos.  |  d.  |   |     P.P. 

0 

I 

9.83  378 

14 

9.96  966 

25 
11 

0.03  034 

9-86413 

12 
12 

60 

59 

9.83  392 

9.96991 

0.03  009 

9.86  401 

2 

9.83  405 
9.83419 

9.97016 
9.97  042 

0.02  984 
0.02  958 

9-86  389 
9.86377 

12 

58 
57 

26 

25 

4 

9-83432 

U 

9.97  067 

25 

25 
26 

0.02  933 

9.86  366 

1 2 

56 

I 

2.6 

2-5 

5 

9.83  446 

14 
13 

13 
14 
13 

13 
14 

9.97  092 

0.02  908 

9-86  354 

12 

55 

5:s 

10.4 
13.0 
15.6 
18.2 
20.8 
23.4 

5-0 

6 

7 

9-83  459 
983  473 

9.97  "8 
9-97  143 

25 

25 
25 
26 

25 

25 
26 

0.02  882 
0.02  857 

9-86  342 
9.86330 

12 
12 

54 
53 

3 

4 
5 
6 

7-5 
1 0.0 
12.5 
15.0 

17-5 
20.0 
22.5 

8 

9.83486 

9.97  168 

0.02  832 

9.86318 

52 

9 
10 

II 

.9.83  500 

9.97  IQ3 

0.02  807 

9.86  306 

II 
12 

12 

5* 
50 

49 

I 

9 

983513 
9.83  527 

9.97  219 

0.02  781 

9.86  295 

9.97  244 

0.02  750 

9.86  283 

12 

9.83  540 

9.97  269 

0.02  731 

9.86271 

12 

48 

^3 

9-83  554 

9.97  295 

25 
11 

0.02  705 

9.86  259 

1 2 

47 

14 

9-83  567 

13 

14 

9.97  320 

0.02  680 

9.86  247 

46 

^S 

9.83  581 

9-97  345 

0.02  655 

9.86  235 

45 

14 

13 

lb 

983  594 

^3 

9-97  37^ 

25 
25 

0.02  629 

9.86  223 

44 

I 

1.4 

1-3 

17 

9.83  608 

14 
13 
13 
14 
13 

9.97  396 

0.02  604 

9.86  211 

I  T 

43 

2 

2.8 

2.6 

i8 

9.83621 

9.97  421 

0.02  579 

9.86  200 

I  2 

42 

3 

4.2 

3-9 

19 
20 

21 

983  634 

9-97  447 

25 
25 
26 

0.02  553 

9.86  188 

12 
12 

T  '? 

4J 
40 

39 

4 

5.6 
7.0 

8.4 
9.8 

5-2 
6-5 
7.8 

9.83  648 

9.97  472 

0.02  528 

9.86  176 

9.83601 

9-97  497 

0.02  503 

9.86  164 

22 

9.83  674 

13 

9.97  523 

0.02  477 

9.86  152 

38 

'J 
8 

9 

9-1 

23 
24 

9.83  688 
9.83  701 

'4 
13 

9-97  548 
9-97  573 

25 

25 

25 
26 

25 

25 
26 

25 
25 

9fi 

0.02452 
0.02  427 

9.86  140 
9.86128 

12 

37 
36 

12.6 

10.4 
11.7 

25 

9-.83  7I5 

H 

9.97  598 

0.02  402 

9.86  116 

T  0 

35 

26 

9.83  728 

M 

9.97  624 

0.02  376 

9.86  104 

12 
12 

34 

27 

9.83  741 

13 

9.97  649 

0.02  351 

9.86092 

33 

12 

11 

28 

9-83  755 

H 

9.97  674 

0.02  326 

9.86  080 

32 

29 

30 

31 

9.83  768 

13 
13 
14 
13 

9.97  700 

0.02  300 

9.86  068 

12 
12 
1 2 

31 
30 

29 

2 
3 
4 

2.4 
3-6 

4.8 

2.2 
3-3 
4-4 

9.83  781 

9-97  725 

0.02  275 

9.86  056 

983  795 

9-97  750 

0.02  250 

9.86  044 

32 

9.83  808 

9.97  776 

25 

0.02  224 

9.86032 

T  -7 

28 

5 

6.0 

5-5 

33 

9.83821 

13 

9.97  801 

0.02  199 

9.86  020 

27 

6 

7.2 

6.6 

34 

9-83  834 

M 

9.97  826 

25 

0.02  174 

9.86  008 

26 

7 

8.4 

7-7 

3S 

9.83  848 

14 

9-97851 

11 

0.02  149 

9.85  996 

25 

8 

9-6 

8.8 

36 

9.83  861 

U 

9.97877 

0.02  123 

9-85  984 

12 

24 

9 

10.8 

9-9 

37 

9.83  874 

13 

9.97  902 

25 

0.02  098 

9.85  972 

23 

3i^ 

9.83  887 

U 

9.97  927 

25 
26 

25 
25 
26 

0.02  073 

9-85  960 

12 
12 
12 

22 

39 
40 

41 

9.83901 

13 
73 

9-97  953 

0.02  047 

9-85  948 

21 
20 

19 

9.83914 

9-97  978 

0.02  022 

9.85  936 

9.83927 

9.98  003 

o.oi  997 

9.85  924 

42 

9.83  940 

M 

9.98  029 

o.oi  971 

9.85912 

18 

43 

9.83  954 

A4 

9.98  054 

25 

O.OI  946 

9-85  900 

17 

13 

13   12 

44 

9.83  967 

U 

9.98079 

25 

O.OI  921 

9.85  888 

16 

26  i 

25   25 

4S 

9.83  9S0 

M 

9.98  104 

11 

25 

O.OI  896 

9.85  S76 

12 

vs 

0 

I.O   I 

.0  1.0 

46 

9-83  993 

13 

9.98  130 

O.OI  870 

9.85  864 

14 

1 

3.0  2 

•9  3-1 

47 

9.84  006 

^3 

9-98  155 

O.OI  843 

9.85851 

13 
12 

13 

2 

5-0  A 

.8  5.2 

48 

9.84020 

M 

9.98  180 

11 
25 
25 

O.OI  820 

9-85  839 

12 

3 

7.0  6 

-7  7-3 

49 
50 

51 

984033 

13 
13 

9.9S  206 

O.OI  794 

9-85  827 

12 
12 

II 
10 

9 

4 

I 

7 

9.0  8 
1 1.0  IC 
13.0  12 

.7  9.4 
.6  11.5 

-5  13.5 

9.84046 

9.98231 

O.OI  769 

9.85  815 

9.84059 

9.98  256 

O.OI  744 

9.85  803 

52 

9.84072 

13 

9.98281 

26 

O.OI  719 

9-85  791 

8 

8 

15.0  14.4  15.0 

S3 

9.84085 

M 

9.98  307 

O.OI  693 

985  779 

7 

Q 

17.0  16.3  17.7 
19.0  18.3  19.8 
21.0  20.2  21.9 

23.0  22.1  24.0 

54 

55 

9.84098 
9.84  112 

14 
13 

9-98  332 
^9.98357 

^5 

25 
26 

O.OI  668 
O.OI  643 

9.85  766 
9-85  754 

12 

1 2 

6 

5 

10 
II 

5^ 

9.84125 

9-98383 

25 

O.OI  617 

9.85  742 

4 

12 

25.0  2^ 

l-.o  — 

57 

9.84  138 

13 

9.98  408 

O.OI  592 

9.85  730 

3 

13 

58 

9.84151 

13 

998433 

25 

O.OI  567 

9.85718 

I- 

2 

59 
60 

9.84  164 

13 
13 

998458 

il 

O.OI  542 

9.85  706 

13 

I 
0 

9.841/7 

9-98  484 

O.OI  516 

9.85  693 

L.  Cos.  1  d. 

L.  Cot.  |c.  d. 

1  L.  Tan. 

L.  Sin.  1  d. 

t 

P.P. 

Aa^ 


70 


' 

L.  Sin. 

d. 

L.  Tan. 

c.  d. 

L.  Cot. 

L.  Cos. 

1  d. 

P.  P. 

0 

I 

9.84177 

13 
13 
13 
13 
13 
13 

9.98484 

25 

25 

26 

o.oi  516 

9.85  693 

12 

60 

59 

9,84  190 

9.98  509 

0.0 1  491 

9.85  681 

2 

9.84  203 

9-98  534 

O.OI  466 

9.85  669 

S8 

26  25  14 

3 

9.84216 

9.98  560 

25 
25 
25 
26 

O.OI  440 

9.85  657 

T  0 

57 

I 

2.6  2.5  14 

4 

9.84  229 

9.98  585 

O.OI  415 

9.85  645 

13 

56 

2 

5.2  5.0  2.8 

S 

9.84  242 

9.98  010 

O.OI  390 

9.85  632 

55 

3 

7.8  7.5  42 

6 

9-^4  255 

998  635 

O.OI  365 

9.85  620 

54 

4 

0.4  lo.o  5.6 

7 

9.84  269 

14 

9.98661 

O.OI  339 

9.85  608 

5S 

5 

3-0  12.5  7.0 

8 

9.84  282 

13 

9.98  686 

25 

O.OI  314 

9-85  596 

52 

6 

5.6  15.0  8.4 

9 
10 

11 

9.84  295 
9.64  308 
9.84321 

13 
13 
13 
13 

9.98  711 
9-98  737 

2^ 
25 
25 
25 
26 

O.OI  289 

9.85  583 

^3 
12 

12 

12 

13 

51 
50 

49 

7  15.2  17.5   9.5 

8  20.8  20.0  II. 2 

9  23.4  22.5  I2.b 

O.OI  263 

9-85  571 

9.98  762 

O.OI  238 

9-85  559 

12 

9-^4  334 

9.98  787 

O.OI  213 

9.85  547 

48 

13 

9-^4  347 

9.98  812 

O.OI  188 

9-85  534 

47 

1.^    19. 

H 

9.84  360 

9.98  838 

25 
25 
25 
26 

O.OI  162 

9.85  522 

12 

13 
12 
1 2 

46 

IS 

9.^4  373 

12 
I  -1 

9.98  863 

O.OI  137 

9.85510 

45 

1.3   1.2 
2.6   2.4 

3-9   3.6 

5.2   4.8 

i6 

9.84  3«5 

9.98  888 

O.OI  112 

9-85  497 

44 

3 

I? 

9.84  398 

I  ■> 

9.98913 

O.OI  087 

9.85  485 

43 

i8 

9.84  41 1 

13 
13 

9-98  939 

25 
25 
26 

25 
25 
25 
26 

O.OI  061 

9-85  473 

42 

5 
6 

I 

(x.S       6.0 

7.8   7.2 

9.1   8.4 

10.4   9.6 

19 
20 

21 

9.84  424 

9-^4  437 
9.84450 

9.98  964 
9.98  989 

O.OI  036 

9.85  460 

13 
12 

12 

13 
12 

12 

41 
40 

39 

O.OI  on 

9.85  448 

9.99015 

0.00985 

9-85  436 

22 

9.84  463 

13 
13 

9.99  040 

0.00  960 

9.85  423 

38 

9 

11.7  10.8 

23 

9.84476 

9.99  065 

0.00  935 

9-85411 

37 

24 

9.84489 
9.84  502 
9.84515 

9.99  090 
9.99  116 
9.99  141 

0.00910 
0.00  884 
0.00  859 

9-85  399 
9-85  i86-. 
9:35-374 

36 

25 
26 

13 

25 

% 

26 
25 
25 

«- 

4 

27 

9.84  528 

^j 

9.99  166 

0.00  834 

9.85361 

13 

ZZ 

13   13 

28 

9.84  540 

13 
13 
13 
13 
13 
13 
12 

13 

9.99  191 

0.00  809 

9-85  349 

32 

29 

30 

31 

9-84  553 

9.99217 

0.00  783 

9-85  337 

13 

12 

13 
12 

13 

31 
30 

29 

26   25 

9.84  566 

9.99  242 

0.00  758 

9-85  324 

0 
I 

I.O    I.O 

3.0   2.9 

9M  579 

9.99  267 

0.00  733 

9.85312 

32 

9.84  592 

9.99  293 

25 

25 

25 
26 

0.00  707 

9.85  299 

28 

2 

5.0   4.8 

Zi 

9.84  605 

9.99318 

0.00  682 

9.85287 

27 

3 

7.0  6.7 

34 

9.84618 

9-99  343 

0.00  657 

9-85  274 

26 

4 

9.0    8.7 

3S 

9.84  630 

9.99  368 

0.00  632 

9.85  262 

12 

25 

I  I.O  10.6 

36 

9.84  643 

9-99  394 

0.00  606 

9-85  250 

24 

I 

13.0  12.5 

37 

9.84  656 

13 

9.99419 

25 

0.00  581 

9-85  237 

13 

23 

15.0  14.4 

38 

9.84  669 

13 

9.99  444 

25 

0.00  556 

9.85  225 

22 

9 

17.0  16.3 
19.0  18.3 
21.0  20.2 

23.0   22.1 
25.0   24.0 

39 
40 

41 
42 

9.84  682 

13 
12 

13 
13 
13 

9.99  469 

26 
25 
25 
25 
26 

0.00  531 

9.85212 

13 
12 

13 
12 
13 

21 
20 

18 

10 
II 
12 
13 

9.84  694 

9.99  495 

0.00  505 

9.85  200 

9.84  707 
9.84  720 

9.99  520 
9-99  545 

0.00  480 
0.00  455 

9.85  187 
9.85  175 

43 

9-84  733 

9-99  570 

0.00  430 

9.85  162 

17 

44 

9-84  745 

13 
13 
13 
12 

9.99  596 

25 
25 
26 

25 
25 
25 
26 

25 
25 

0.00  404 

9.85  150 

13 
12 

13 
12 

16 

45 

9.84  758 

9.99  521 

0.00  379 

985  137 

15 

12   12 

46 

9.84771 

9.99  646 

0.00  354 

9-85  125 

14 

26   25 

47 

9.84  784 

9.99  672 

0.00  328 

9.85  112 

13 

0 

I.I    I.O 

48 

9.84  796 

9.99  697 

0.00  303 

9.85  100 

12 

I 

3-2   3-1 

49 
50 

51 

9.84  809 

13 
13 
12 
13 

9.99  722 

0.00  278 

9.85  087 

13 
12 

13 
1 2 

II 
10 

9 

2 

3 
4 

5 
6 

5-4   5-2 
7-6   7-3 
9-8   9-4 

9.84  822 

9-99  747 

0.00  253 

9.85  074 

9.84835 

9-99  773 

0.00  227 

9.85  062 

52 

9.84847 

9.99  798 

0.00  202 

9.85  049 

8 

11.9  11.5 

53 

9.84  860 

9.99  823 

0.00  177 

9.85  037 

7 

14.1  13-5 

54 

9.84873 

13 
12 

9.99848 

25 
26 

0.00  152 

9.85  024 

13 
12 

6 

16.2  15.6 
18.4  17.7 
20.6  19.8 

22.8  21.9 

24.9  24.0 

55 

9.84  885 

9.99  874 

0.00  126 

9.85  012 

5 

9 

56 

9.84898 
9.84  91 1 

13 
13 

9.99  899 
9.99  924 

25 

25 

"I 

0.00  10 1 
0.00  076 

9.84  999 
9.84  986 

13 

4 
3 

10 
II 

5« 

9.84  923 

13 
13 

9.99  949 

0.00051 

9.84  974 

13 

12 

2 

12 

59 
60 

9.84  936 

9-99  975 

25 

0.00  025 

9.84961 

I 
0 

9.84  949 

0.00  000 

0.00  000 

9.84  949 

L.  Cos. 

d. 

L.  Cot.  |c.  d. 

L.  Tan. 

L.  Sin. 

d.  ' 

P.P.  ; 

46' 


n 


III. 

NATURAL 
TRIGONOMETRIC    FUNCTIONS 

FOR   EACH    MINUTE. 


72 

, 

0° 

1 

^ 

1 

t 

N.Sin. 

|N.Tan.|N.Cot. 

N.Cos.l 

0 

I 

.00000 

.00000 

00 

1. 0000 

60 

59 

029 

029 

3437-7 

000 

2 

058 

058 

1718.9 

000 

58 

3 

087 

087 

"45-9 

000 

57 

4 

116 

116 

85944 

000 

56 

5 

.00145 

.00145 

687.55 

1. 0000 

55 

6 

175 

175 

572.96 

000 

54 

7 

204 

204 

491. II 

000 

53 

8 

233 

233 

429.72 

000 

52 

9 
10 

II 

262 

262 

381.97 

000 

51 
50 

49 

.00291 

.00291 

343.77 

1. 0000 

320 

320 

312.52 

-99999 

12 

349 

349 

286.48 

999 

48 

13 

37« 

378 

264.44 

999 

47 

14 

407 

407 

245-55 

999 

46 

IS 

.00436 

.00436 

229.18 

.99999 

4S 

16 

465 

465 

214.86 

999 

44 

■ 

17 

495 

495 

202.22 

999 

43 

18 

524 

524 

190.98 

999 

42 

19 

20 

21 

553 

553 

180.93 

998 

41 
40 

39 

.00582 

.00582 

171.89 

.99998 

611 

611 

163.70 

998 

22 

640 

640 

156.26 

998 

38 

23 

669 

669 

149.47 

998 

37 

24 

698 

698 

143.24 

998 

36 

25 

.00727 

.00727 

137-51 

■99997 

3S 

26 

756 

756 

132.22 

997 

34 

27 

785 

785 

127.32 

997 
997 

33 

28 

814 

81S 

122.77 

32 

29 

30 

31 

844 

844 

118.54 

996 

31 
30 

29 

.00873 

.00873 

"4-59 

.99996 

902 

902 

110.89 

996 

32 

931 

931 

107.43 

996 

28 

33 

9bo 

960 

104.17 

995 

27 

34 

.00989 

.00989 

lOI.II 

995 

26 

35 

.01018 

.01018 

98.218 

.99995 

25 

36 

047 

047 

95.489 

995 

24 

37 

076 

076 

92.908 

994 

23 

38 

105 

105 

90.463 

994 

22 

39 
40 

41 

134 

135 

88.144 

994 

21 
20 

19 

.01164 

.01164 

85.940 

.99993 

193 

193 

83.844 

993 

42 

222 

222 

81.847 

993 

18 

43 

251 

251 

79.943 

992 

17 

44 

280 

280 

78,126 

992 

16 

4S 

.01309 

.01309 

76.390 

.99991 

15 

46 

33^ 

338 

74.729 

991 

11,4 

47 

367 

367 

73-139 

991 

13 

48 

396 

396 

71.615 

990 

12 

49 
50 

SI 

425 

425 

70.153 

990 

II 
10 

9 

.01454 

•01455 

68.750 

.99989 

483 

484 

67.402 

989 

S2 

513 

513 

66.105 

989 

8 

53 

542 

542 

64.858 

988 

7 

S4 

571 

571 

63-657 

988 

6 

ss 

.01600 

.01600 

62.499 

-99987- 

-^ 

56 

629 

629 

61.383 

987 

4 

S7 

658 

658 

60.306 

986 

3 

sB 

687 

687 

S9.266 

986 

2 

59 
60 

716 

716 

58.261 

985 

I 
0 

•01745 

.01746 

57.290 

.99985 

N.  Cos. 

N.Cot.lN.Tan. 

N.Sin.j  '  i 

^ 

1 

0 

'  iN.Sin. 

N.Tan.|N.Cot. 

N.Cos 

0 

I 

•01745 

.01746 

57-290 

.99985 

60 

S9 

774 

775 

56.351 

984 

2 

803 

804 

55^442 

984 

S8 

3 

832 

833 

54^561 

983 

57 

4 

862 

862 

53-709 

983 

S6 

5 

.01891 

.01S91 

52.882 

.99982 

SS 

6 

920 

920 

52.081 

982 

54 

7 

949 

949 

51^303 

981 

S3 

8 

.01978 

.01978 

50.549 

980 

S2. 

9 
10 

II 

.02007 

.02007 

49.816 

980 

51 
50 

49 

.02036 

.02036 

49.104 

.99979 

065 

066 

48.412 

979 

12 

094 

095 

47.740 

978 

48 

13 

123 

124 

47-085 

977 

47 

14 

152 

153 

46.449 

977 

46 

15 

.02181 

.02182 

45-829 

.99976 

4S 

16 

211 

211 

45.226 

976 

44 

17 

240 

240 

44.639 

97S 

43 

18 

269 

269 

44.066 

974 

42 

19 
20 

21 

298 

298 

43-508 

974 

41 
40 

39 

-02327 

.02328 

42.964 

•99973 

356 

357 

42.433 

972 

22 

385 

386 

41.916 

972 

38 

23 

414 

415 

41.411 

971 

37 

24 

443 

444 

40.917 

970 

36 

25 

.02472 

-02473 

40.436 

.99969 

3S 

26 

501 

502 

39.965 

969 

34  ■ 

27 

530 

531 

39-506 

968 

33 

28 

560 

560 

39.057 

967 

32 

29 
30 

31 

589 

589 

38.618 

966 

31 
30 

29 

.02618 

.02619 

38.188 

.99966 

647 

648 

37.769 

965 

32 

676 

677 

37.358 

964 

28 

33 

705 

706 

36.956 

963 

27 

34 

734 

735 

36.563 

963 

26 

35 

.02763 

.02764 

36.178 

.99962 

2S 

36 

792 

793 

35.801 

961 

24 

37 

821 

822 

35-431 

960 

23 

3^ 

850 

851 

35-070 

959 

22 

39 
40 

41 

879 

881 

34.715 

959 

21 

20 

19 

.02908 

.02910 

34.368 

•99958 

938 

939 

34.027 

957 

42 

967 

968 

33.694 

956 

18 

43 

.02996 

.02997 

33.366 

955 

17 

44 

.03025 

.03026 

33.045 

954 

16 

45 

.03054 

-03055 

32.730 

•99953- 

IS 

46 

083 

084 

32.421 

952 

14 

47 

112 

114 

32.118 

952 

13 

48 

141 

143 

31.821 

951 

12 

49 
50 

SI 

170 

172 

31.528 

950 

II 
10 

9 

.03199 

.03201 

31.242 

.99949 

228 

230 

30.960 

948 

S2 

257 

259 

.30.683 

947 

8 

53 

286 

288 

30.412 

946 

7 

S4 

316 

317 

30.145 

•945 

6 

SS 

■03345 

.03346 

29.882 

.99944 

5 

56 

374 

376 

29.024 

943 

4 

57 

403 

405 

29.371 

942 

3 

S8 

432 

434 

29.122 

941 

2 

59 
60 

461 

463 

28.877 

940 

0 

.03490 

-03492 

28.636 

•99939 

1 

N.Cos.|N.Cot. 

N.Tan. 

N.  Sin. 

' 

89° 


88' 


2° 

A 

1 

N.  Sin. 

N. Tan. In.  Cot. 

N.Cos. 

0 

I 

.03490 
519 

.03492 

28.636 
•399 

.99939 
938 

60 

59 

5ii 

2 

54Ji 

550 

28.166 

937 

58 

3 

577 

579 

27-93r 

936 

"57 

4 

606 

609 

.712 

935 

S6 

5 

•03635 

.03638 

27,490 

•99934 

55 

0 

bb4 

667 

.271 

933 

54 

7 

693 

696 

27.057 

932 

S3 

8 

723 

725 

26.845 

931 

52 

9 
10 

11 

752 

754 

.637 

930 

49 

.03781 

.03783 

26.432 
.230 

•99929 
927 

810 

812 

12 

«39 

842 

26.031 

926 

48 

'3 

8b8 

871 

25-835 

925 

47 

'4 

897 

900 

.642 

924 

46 

'5 

.03926 

.03929 

25452 

•99923 

45 

lb 

955 

958 

.264 

922 

44 

17 

.03984 

.03987 

25.080 

921 

43 

18 

.04013 

.04016 

24.898 

919 

42 

19 
20 
21 

042 

046 

.719 

918 

41 
40 

39 

.04071 

.04075 

24.542 

.99917 

100 

104 

.368 

916 

22 

129 

^33 

.196 

915 

?>^ 

23 

159 

162 

24.02b 

913 

37 

I24 

188 

191 

23.859 

912 

36 

25 

.04217 

.04220 

23^695 

.99911 

35 

2b 

246 

250 

•532 

910 

34 

l'2 

275 

279 

•372 

909 

?>?> 

,28 

304 

308 

.214 

907 

32 

29 

30 

31 

33Z 

337 

23.058 

906 

31 
30 

29 

.043b2 

.04366 

22,904 

.99905 

391 

395 

•752^ 

904 

32 

420 

424 

.602 

902 

28 

33 

449 

454 

•454 

901 

27 

34 

478 

483 

.308 

900 

26 

35 

.04507 

.04512 

22.164 

.99898 

2S 

3b 

536 

541 

22.022 

897 

24 

37 

565 

570 

21.881 

896- 

23 

3« 

594 

599 

•743 

894 

22 

39 
40 

41 

623 

628 

.606 

893 

21 

20 

19 

.04653 

.04658 

21.470 

.99892 

682 

687 

•337 

890 

42 

711 

716 

.205 

889 

18 

43 

749 

745 

21.075 

888 

17 

44 

769 

774 

20.946 

886 

16 

45 

.04798 

.04803 

20.819 

•9988s 

15 

4b 

827 

833 

.693 

883 

H 

47 

856 

862 

.569 

882 

13 

48 

885 

891 

.446 

881 

12 

49 
50 

51 

914 

920 

•325 !  879 

II 
10 

9 

•04943 

.04949 
.04978 

20.206 1 .99878 

.04972 

20.087 

87b 

52 

.05001 

.05007 

19.970 

87.5 

8 

53 

030 

037 

.855 

873 

7 

54 

059 

066 

•740 

872 

6 

55 

.05088 

•05095 

i9-627 

.99870 

5 

5t> 

117. 

124 

.516 

869 

4 

^57 

146 1   153 

.405 

867 

3 

5^ 

175 

182 

.296 

866 

2 

59 
60 

205 

212 

.188 

864 

ii 

.05234  1  .05241 

19.081 

.99863 

.N.Cos.|n.  Cot.|N.Tan.|  N.  Sin.)  '  J\\ 

8 

T 

) 

^ 

3° 

73 

/ 

N.  Sin.  N. Tan. 

N.  Cot.  N.Cos. 

0 

.05234 

.05241 

19.081 

.99863 

60 

I 

263 

270 

18.976 

861 

59 

2 

292 

299 

.871 

860 

S8 

3 

321 

328 

.768 

858 

57 

4 

350 

357 

.666 

857 

56 

S 

•05379 

.05387 

18.564 

•99855 

55 

6 

408 

416 

.464 

854 

54 

7 

437 

445 

.366 

852 

53 

8 

466 

474 

.268 

851 

52 

9 
10 

II 

495 

503 

.171 

^849 

.99847 
846 

5' 
50 

49 

05524 

•05533 

18.075 
17.980 

553 

562 

12 

582 

591 

.886 

844 

48 

13 

611 

620 

.793 

842 

47 

14 

640 

649 

.702 

841 

46 

15 

.05669 

•05678 

17.611 

•99839 

45 

16 

698 

708 

.521 

838 

44 

17 

727 

737 

.431 

836 

43 

18 

7S6 

766 

.343 

834 

42 

19 

785 

795 

.256 

833 

41 

20 

21 

.05814 

.05824 

17.169 
17.084 

.99831 

40 

39 

844 

854 

829 

22 

873 

883 

16.999 

827 

38 

23 

902 

912 

•915 

826 

37 

24 

931 

941 

.832 

824 

36 

25 

.05960 

•05970 

16.750 

.99822 

35 

26 

.05989. 

.05999 

.668 

821 

34 

27 

.06018 

.06029 

•587 

819 

33 

28 

047 

058 

•507 

817 

32 

29 
30 

31 

076 

087 

.428 

815 

31 
30 

29 

.06105 

.06  u6 

16.350 

.99813 

134 

145 

.272 

812 

32 

163 

175 

.195 

810 

28 

33 

192 

204 

.119 

808 

27 

34 

221 

233 

16.043 

806 

26. 

35 

.06250 

.06262 

15.969 

.99804 

25 

36 

279 

291 

.895 

803 

24 

37 

308 

321 

.821 

801 

23 

38 

337 

350 

.748 

799 

22 

39 
40 

41 

366 

379 

.676 

797 

19 

.06395 

.06408 

15.605 

.99795 

424 

438 

•534 

793 

42 

453 

467 

•464 

792 

18 

43 

482 

-  496 

.394 

790 

17 

44 

5" 

525 

•325 

788 

16 

4S 

.06540 

•06554 

15^257 

.99786 

IS 

46 

569 

584 

.189 

784 

14 

47 

598 

613 

.122 

782 

13 

48 

b27 

642 

15056 

780 

12 

49 
50 
51 

656 

671 

14.990 

778 

11 
iO 

9 

.06685 

.06700 

14.924 

.99776 

7»4 

•  730 

.860 

774 

52 

743 

759 

.795 

772 

8 

53 

773 

788 

.732 

770 

7 

54 

802 

817 

.669 

768 

6 

5S 

.06831 

.06847 

14.606 

.99766 

5 

56 

860 

876 

•544 

764 

4 

57 

889 

90s 

.482 

762 

3 

■.58 

918 

934 

.421 

760 

2 

59 
60 

947 

963 

.361 

758 

0 

.06976 

.06993 

14.301 

.99756 

N.Cos. 

N.Cot.  N.Tan.lN.Sin. 

1 

86° 


74 

t 

4° 

w\ 

/ 

N.Sin.jN.Tan. 

N.  Cot. 

N.Cos 

0 

I 

.06976 

.06993 

14.301 

•99756 

60 

59 

.07005 

.07022 

.241 

754 

2 

034 

051 

.182 

752 

58 

3 

063 

080 

.124 

750 

57 

4 

092 

no 

.065 

748 

56 

5 

.07121 

.07139 

14.008 

.99746 

55 

b 

150 

168 

13-951 

744 

54 

7 

179 

197 

.894 

742 

53 

8 

208 

227 

.838 

740 

52 

9 
lO 

237 

256 

.782 

738 

51 
50 

49 

.07266 

.07285 

13.727 

•99736 

295 

3'4 

.672 

734 

12 

324 

344 

.6.7 

731 

48 

13 

353 

373 

.563 

729 

47 

H 

382 

402 

.510 

727 

46 

'5 

.07411 

.07431 

13457 

.99725 

45 

10 

440 

461 

.404 

723 

44 

17 

469 

490 

•352 

721 

43 

i8 

498 

519 

.300 

719 

42 

19 
20 

21 

527 

548 

.248 

716 

41 
40 

39 

.07556 

.07578 

13-197 

.99714 

585 

607 

.146 

712 

22 

614 

636 

.096 

710 

38 

23 

643 

665 

13.046 

708 

37 

24 

672 

695 

12.996 

705 

36 

2.S 

.07701 

.07724 

12.947 

•99703 

35 

26 

730 

753 

.898 

701 

34 

27 

759 

782 

.850 

699 

33 

28 

788 

812 

.801 

696 

32 

29 

30 

31 

817 

841 

•754 

694 

31- 
30 

29 

.07846 

.07870 

12.706 

.99692 

875 

899 

•659 

689 

32 

904 

929 

.612 

687 

28 

33 

933 

958 

.566 

685 

27 

34 

962 

.07987 

.520 

683 

26 

3S 

.07991 

.08017 

12.474 

.99680 

25 

3^ 

.08020 

046 

.429 

678 

24 

37 

049 

075 

.384 

676 

23 

3« 

078 

104 

.339 

673 

22 

39 
40 

41 

107 

134 

.295 

671 

21 
20 

19 

.08136 

.08163 

12.251 

.99668 

165 

192 

.207 

666 

42 

194 

221 

.163 

664 

18 

43 

223 

251 

.129 

66 1 

17 

44 

252 

280 

.077 

659 

16 

45 

.08281 

.08309 

12.035 

•99657 

15 

4b 

310 

339 

11.992 

654 

14 

47 

339 

368 

.950 

652 

13 

48 

368 

397 

.909 

649 

12 

49 
50 

SI 

397 

427 

.867 

647 

11 
10 

9 

.08426 

.08456 

11.826 

.99644 

455 

485 

.785 

642 

52 

484 

5H 

•745 

639 

8 

53 

513 

544 

.705 

637 

7 

S4 

542 

573 

.664 

635 

6 

55 

.08571 

.08602 

11.625 

.99632 

5 

56 

600 

632 

•585 

630 

4 

57 

629 

661 

.546 

627 

3 

58 

658 

690 

.507 

625 

2 

59 
60 

687 

720 

.468 

622 

1 
0 

.08716 

.08749 

11.430 

.99619 

N.  Cos.  N.  Cot. 

N.Tan.iN.Sin.| 

.n 

1 

N.Sin 

iN.Tan.  N.Cot. 

1  N.Cos 

0 

I 

.08716 

.08749 

11.430 

.99619 

60 

59 

745 

778 

.392 

617 

2 

774 

807 

•354 

614 

58 

3 

803 

837 

.316 

612 

57 

4 

831 

866 

•279 

609 

S6 

5 

.08860 

.0S89S 

11.242 

.99607 

55 

6 

889 

925 

.205 

604 

54 

7 

918 

954 

.168 

602 

S3' 

8 

947 

.08983 

.132 

599 

52 

9 
10 

11 

.08976 

.09013 

.095 

5,6 

51 
50 

49 

.09005 

.09042 

11.059 

•99594 

034 

071 

11.024 

591 

12 

063 

lOI 

10.988 

588 

48 

13 

092 

130 

•953 

586 

47 

14 

121 

159 

.918 

583 

46 

15 

.09150 

.09189 

10.883 

•99580 

45 

16 

179 

218 

.848 

578 

44 

17 

208 

247 

.814 

575 

43 

18 

237 

277 

.780 

572 

42 

19 
20 

21 

266 

306 

.746 

570 

41 
40 

.39 

09295 

•09335 

10.712 

.99567 

'  324 

365 

.678 

564 

22 

353 

394 

.641 

562 

38 

23 

3^^ 

423 

.612 

559 

37 

24 

411 

453 

.579 

556 

36 

25 

.09440 

.09482 

10.546 

•99553 

35 

26 

469 

5" 

.5M 

551 

34 

27 

498 

541 

.481 

548 

33 

28 

52? 

570 

.449 

545 

32 

29 
30 

31 

556 

600 

.417 

542 

31 
30 

29 

•09585 

.09629 

10.385 

.99540 

614 

658 

.354 

537 

32 

642 

688 

.322 

534 

28 

33 

671 

717 

.291 

531 

27 

34 

700 

746 

.260 

528 

26 

35 

.09729 

.09776 

10.229 

.99526 

25 

36 

758 

805 

.199 

523 

24 

37 

787 

834 

.168 

520 

23 

38 

816 

864 

.1.S8 

517 

22 

39 
40 

41 

845 

893 

.108 

5H 

21 
20 

19 

.09874 

.09923 

10.078 

•995" 

903 

952 

.048 

508 

42 

932 

.09981 

10.019 

506 

18 

43 

961 

.10011 

9.9893 

503 

17 

44 

.09990 

040 

.9601 

500 

16 

45 

.10019 

.10069 

9.9310 

.99497 

15 

46 

048 

099 

.9021 

494 

14 

47 

077 

128 

.8734 

491 

13 

48 

106 

158 

.8448 

4S8 

12 

49 
50 

51 

135 

187 

.8164 

485 

11 

io 

9 

.10164 

.10216 

9.78^2 

.99482 

192 

246 

.7601 

479 

52 

221 

275 

.7322 

476 

8 

53 

250 

305 

.7044 

473 

7 

54 

279 

334 

.6768 

470 

6 

55 

.10308 

.10363 

9-6493 

.99467 

S 

56 

337 

393 

.6220 

464 

4 

57 

366 

422 

.5949 

461 

3 

58 

•395 

452 

•5679 

458 

2 

55 
60 

424 

481 

.5411 

455 

1 
0 

.10453 

.10510 

9.5144 

.99452 

C_. 

N.Cos.'N.  Cot.JN.Tan.  N.  Sin.| 

/ 

86' 


84' 


6° 

•* 

T 

75 

/ 

N.Sin. 

N.Tan.jN.Cot. 

N.Cos. 

/ 

N.Sin. 

N.Tan.  N.  Cot. 

N.Cos. 

! 

0 

I 

.10453 
482 

.10510 

9-5 '44 

•99452 

60 

S9 

0 

.12187 

.12278 
308 

8.1443 
.1248 

•99255 
251 

60 

59 

540 

.4878 

449 

216 

2 

511 

569 

.4614 

446 

58 

2 

245 

338 

•'Sl^ 

248 

58 

3 

540 

599 

.4352 

443 

57 

3 

274 

367 

.0860 

244 

57 

4 

569 

628 

.4090 

440 

S6 

4 

302 

397 

.0667 

240 

56 

5 

.10597 

.10657 

9.3831 

•99437 

ss 

5 

.12331 

.12426 

8.0476 

•99237 

55 

6 

626 

687 

•3572 

434 

54 

b 

360 

456 

.0285 

233 

54 

7 

6SS 

716 

•3315 

431 

S3 

7 

389 

485 

8.0095 

230 

53 

8 

684 

746 

.3060 

428 

52 

8 

418 

515 

7.990b 

226 

52 

9 
10 

II 

713 

775 

.2806 

424 

51 
50 

49 

9 
iO 

II 

447 

544 

.9718 

222 
.99219 

51 
50 

49 

.10742 

.10805 

834 

9.2553 

.99421 

.12476 

•12574 

79530 

771 

.2302 

418 

504 

603 

•9344 

215 

12 

800 

863 

.2052 

415 

48 

12 

533 

633 

.9158 

211 

48 

13 

829 

893 

.1803 

412 

47 

13 

562 

662 

.8973 

208 

47 

H 

858 

922 

.1555 

409 

46 

14 

591 

692 

•8789 

204 

46 

^S 

.10887 

.10952 

9.1309 

.99406 

4S 

rs 

.12620 

.12722 

7.8606 

.99200 

45 

lb 

916 

.10981 

.1065 

402 

44 

16 

649 

751 

.8424 

197 

44 

^7 

94^ 

.11011 

.0821 

399 

43 

'7 

678 

781 

:8243 

193 

43 

i8 

.10973 

040 

•0579 

396 

42 

18 

706 

810 

.8062 

189 

42 

'19 
20 

21 

.11002 

070 

.0338 

393 

41 
40 

39 

19 
20 

21 

735 

840 

.7882 

186. 

41 
40 

19 

.11031 

.11099 
128 

9.0098 

.99390 

.12764 

.12869 

7.7704 

.99182 

obo 

8.9860 

386 

793 

899 

•7525 

178 

22 

089 

158 

.9623 

383 

38 

22 

822 

929 

•7348 

17^ 

38 

23 

118 

187 

•9387 

380 

37 

23 

851 

958 

.7171 

171 

37 

24 

147 

217 

.9152 

377 

36 

24 

880 

.12988 

.6996 

167 

36 

25 

.11176 

.11246 

8.8919 

•99374 

3S 

2S 

.12908 

.13017 

7.6821 

.99163 

3S 

2b 

205 

276 

.8686. 

370 

34 

26 

937 

047 

.6647 

160 

34 

27 

234 

305 

•8455 

367 

33 

27 

966 

076 

.6473 

IS6 

33 

28 

263 

335 

•8225 

•  364 

.32 

28 

•12995 

106 

•6301 

152 

32 

29 

30 

31 

291 

364 

.7996 

360 

31 
30 

29 

29 

30 

31 

.13024 

136 

.6129 

148 

31 
30 

29 

.11320 

•I  1394 

8.7769 

.99357 

•13053 

•13165 

7.5958 

.99144 

349 

423 

.7542 

354 

081 

195 

.5787 

141 

32 

37« 

452 

•7317 

351 

28 

32 

no 

224 

.5618 

137 

28 

33 

407 

482 

.7093 

347 

27 

33 

139 

254 

•5449 

^33 

27 

34 

436 

5" 

.6870 

344 

26 

34 

168 

284 

.5281 

129 

26 

35 

.11465- 

.11541 

8.6648 

.99341 

25 

35 

•13197 

'^3313 

7^5"3 

•99125 

2S 

3^ 

494 

570 

.6427 

337 

24 

36 

226 

343 

.4947 

122 

24 

37 

523 

600 

.6208 

334 

23 

37 

254 

372 

.4781 

118 

23 

3ii 

552 

629 

•5989 

33^ 

22 

38 

283 

402 

.4615 

114 

22 

39 
40 

41 

580 

659 

•5772 

327 

21 
20 

19 

39 
40 

41 

312 

432 

•445' 

no 

21 

20 

19 

.11609 

.11688 

8-5555 

.99324 

.13341 

.13461 

7.4287 

.99106 

638 

718 

.5340 

320 

370 

491 

.4124 

102 

42 

667 

747 

.5126 

317 

18 

42 

399 

521 

.3962 

098 

18 

43 

696 

777 

.4913 

314 

'7 

43 

427 

550 

.3800 

094 

17 

44 

725 

806 

.4701 

310 

16 

44 

456 

580 

•3639 

091 

16 

45 

•"754 

.11836 

8,4490 

•99307 

IS 

4S 

.13485 

.13609 

7^3479 

.99087 

IS 

4b 

.7«3 

865 

.4280 

303 

14 

46 

514 

639 

.33 '9 

0S3 

14 

^Z 

812 

895 

.4071 

300 

13 

47 

543 

669 

.3160 

079 

13 

48 

840 

924 

•3863 

297 

12 

48 

572 

698 

.3002 

075 

12 

49 
50 

51 

869 

954 

.3656 

293 

10 

9 

49 
50 

SI 

600 

728 

.2844 

071 

11 
10 

9 

.11898 

.11983 

8.34I0 

.99290 

.13629 

•13758 

7.2687 

.99067 

927 

.12013 

•3245 

286 

658 

787 

.2531 

063 

52. 

956 

042 

.3041 

283 

8 

S2 

687 

817 

.2375 

059 

8 

53 

.11985 

072 

.2838 

279 

7 

53 

716 

846 

.2220 

055 

7 

54 

.12014 

101 

.2636 

276 

6 

54 

744 

876 

.2066 

051 

6 

55 

.12043 

.12131 

8.2434 

.99272 

S 

SS 

.13773 

.13906 

7.1912 

•99047 

S 

5^ 

071 

160 

.2234 

269 

4 

56 

802 

935 

.1759 

043 

4 

^l 

100 

190 

.2035 

265 

3 

S7 

831 

965 

.1607 

039 

3 

5^ 

129 

219 

•1837 

262 

2 

S8 

860 

.1399? 

.1455 

03? 

2 

59 
60 

158 

249 

.1640 

258 

I 
0 

59 
60 

889 

.14024 

.1304 

031 

I 
0 

.12187 

.12278 

8.1443 

•99255 

.13917 

.14054 

7."54 

.99027 

N.Cos. 

N.  Cot. 

N.Tan. 

N.Sin. 

' 



N.Cos. 

N.Cot.lN.Tan. 

N.Sin.l  '  || 

83° 


82° 


76 

8° 

r 

N.  Sin. 

N.TanJN.Cot. 

N.Cos. 

0 

I 

.13917 

.14054 

7-1154 

.99027 

60 

59 

946 

084 

.1004 

023 

2 

•1397? 

113 

.0855 

019 

58 

3 

.14004 

143 

.0706 

015 

57 

4 

033 

173 

.0558 

on 

56 

S 

,14061 

.14202 

7.0410 

.99006 

55 

6 

090 

232 

,0264 

.99002 

54 

7 

"3 

262 

7.0117 

.98998 

53 

8 

148 

291 

6.9972 

994 

52 

9 
10 

II 

177 

321 

•9827 

990 

51 
50 

49 

.14205 

.1435' 

6.9682 

.98986 

234 

381 

.9538 

982 

12 

263 

410 

•9395 

978 

48 

13 

292 

440 

.9252 

973 

47 

H 

320 

470 

.9110 

969 

46 

IS 

•14349 

.14499 

6.8969 

.98965 

45 

i6 

378 

529 

.8828 

961 

44 

17 

407. 

559 

.8687 

957 

43 

18 

436 

588 

.8548 

953 

42 

19 
20 

21 

464 

618 

.8408 

948 

41 
40 

39 

.14493 

.14648 

6.8269 

•98944 

522 

678 

.8131 

940 

22 

551 

707 

.7994 

936 

38 

23 

580 

737 

.7856 

931 

37 

24 

608 

767 

.7720 

927 

36 

2S 

.14637 

.14796 

6.7584 

•98923 

35 

26 

bb^ 

826 

.7448 

919 

34 

27 

695 

It 

.7313 

914 

ZZ 

28 

723 

886 

.7179 

910 

32 

29 

30 

SI 

752 

915 

•7045 

906 

31 
30 

29 

.14781 

.14945 

6.6912 

.98902 

810 

•14975 

.6779 

897 

32 

838 

.15005 

.6646 

893 

28 

33 

867 

034 

.6514 

889 

27 

34 

896 

064 

.6383 

884 

26 

3S 

.14925 

.15094 

6.6252 

.98880 

25 

36 

954 

124 

.6122 

876 

24 

37 

.14982 

153 

.5992 

871 

23 

38 

.15011 

183 

.5863 

867 

22 

39 
40 

41 

040 

213 

.5734 

863 

21 
20 

19 

.15069 

•15243 

6.5606 

.98858 

097 

272 

•5478 

854 

42 

126 

302 

.5350 

849 

18 

43 

155 

332 

.5223 

845 

17 

44 

184 

362 

.5097 

841 

16 

4S 

.15212 

•15391 

6.4971 

.98836 

15 

46 

241 

421 

.4846 

832 

14 

47 

270 

451 

.4721 

827 

13 

48 

299 

481 

.4596 

823 

12 

49 
50 

SI 

327 

511 

.4472 

8i8 

II 
10 

9 

■  '5356 

•15540 

6.4348 

.98814 

385 

570 

.4225 

809 

S2 

414 

600 

.4103 

80s 

8 

53 

442 

630 

.3980 

800 

7 

S4 

471 

660 

•3859 

796 

6 

ss 

.15500 

.15689 

6.3737 

.98791 

5 

56 

529 

719 

•3617 

787 

4 

S7 

557 

749 

.3496 

782 

3 

S8 

586 

779 

•3376 

778 

2 

59 
60 

615 

809 

.3257 

773 

I 
0 

•15643 

•  15838 

6.3138 

.98769 

N.Cos. 

N.Cot.lN.Tan. 

N.  Sin. 

/ 

' 

N.Sin. 

N.Tan.  N.Cot. 

N.Cos. 

0 

I 

.15643 

.15838 

6.3138 

.98769 

60 

S9 

672 

868 

.3019 

764 

2 

701 

898 

.2901 

760 

S8 

3 

730 

928 

•2783 

755 

57 

4 

758 

958 

.2666 

75» 

S6 

5 

.15787 

.15988 

6.2549 

.98746 

SS 

6 

816 

.16017 

•2432 

741 

54 

7 

845 

047 

.2316 

737 

S3 

8 

873 

077 

.2200 

732 

S2 

9 
10 

II 

902 

107 

.2085 

728 

51 
50 

49 

.15931 

.16137 

6.1970 

•98723 

959 

167 

.1856 

718 

12 

.15988 

196 

.1742 

714 

48 

13 

.16017 

226 

.1628 

709 

47 

14 

046 

256 

.1515 

704 

46 

IS 

.16074 

.16286 

6.1402 

.98700 

45 

16 

103 

316 

.1290 

695 

44 

17 

132 

346 

.1178 

690 

4S 

18 

160 

376 

.1066 

686 

42 

19 

20 

21 

189 

405 

•0955 

681 

41 
40 

39 

.16218 

.16435 

6.0844 

.98676 

246 

465 

.0734 

671 

22 

275 

495 

.0624 

667 

38 

23 

304 

525 

.0514 

662 

37 

24 

zzz 

5SS 

•0405  i   657 

S6 

25 

.16361 

•16585 

6.0296 

.98652 

35 

26 

390 

615 

.0188 

648 

34 

27 

419 

645 

6.0080 

643 

33 

28 

447 

674 

5^9972 

638 

32 

29 
30 

31 

476 

704 

.9865 

633 

31 
30 

29 

.16505 

•16734 

5-9758 

.98629 

533 

764 

.9651 

624 

32 

562 

794 

.9545 

619 

28 

33 

591 

824 

•9439 

614 

27 

34 

620 

854 

.9333 

609 

26 

3S 

.16648 

.16884 

5.9228 

.98604 

25 

36 

677 

914 

.9124 

600 

24 

37 

706 

944 

.9019 

595 

23 

38 

734 

.16974 

.8915 

590 

22 

39 
40 

41 

763 

.17004 

.8811 

585 

21 
20 

19 

.16792 

.17033 

5.8708  1  .98580 

820 

063 

•8605  1   575 

42 

849 

093 

.8502    570 

18 

43 

878 

123 

.8400 1   565 

17 

44 

906 

I  S3 

.8298  i   561 

16 

45 

.16935 

.17183 

58197 

.98556 

15 

46 

964 

213 

.8095 

551 

14 

47 

.16992 

243 

.7994 

546 

13 

48 

.17021 

273 

.7894 

541 

12 

49 
50 

SI 

050 

303 

.7794 

536 

II 
10  1 

9 

.17078 

•17333 

5. 7594}. 985  3 1 

107 

363 

.7594  =   526 

52 

1.36 

393 

.7495    521 

8 

53 

164 

42-3 

.7396 

516 

7 

S4 

193 

453 

.7297 

511 

6 

SS 

.17222 

•17483 

57199 

.98506 

5 

56 

250 

513 

.7101 

501 

4 

S7 

279 

543 

.7004 

496 

3 

S8 

308 

573 

.6906 

491 

2  i 

59 
60 

336 

603 

.6809 

486 

0 

.17365 

•17633 

5.6713 

.98481 

i 

N.Cos. 

N.Cct.JN.Tan. 

N.Sin. 

'J 

8r 


80' 


10' 


» 

N.  Sin 

IN.Tan.jN.Cot 

N.Cos 

0 

I 

.17365 

•17633 
663 

5^67i3 
.6617 

.98481 

60 

59 

393 

476 

2 

422 

693 

.6521 

471 

58 

3 

451 

723 

.6425 

466 

57 

4 

•  479 

753 

.6329 

461 

S6 

5 

.17508 

•17783 

5^6234 

•98455 

55 

b 

537 

813 

.6140 

450 

54 

7 

565 

843 

.6045 

445 

53 

8 

594 

873 

.5951 

440 

52 

9 
lO 

II 

623 

903 

•5857 

435 

51 
50 

49 

.17651 

.17933 

5.5764 

.98430 

680 

963 

•5671 

425 

12 

708 

•17993 

.5578 

420 

48 

13 

737 

.18023 

•5485 

414 

47 

H 

766 

053 

.5393 

409 

46 

15 

.17794 

.18083 

55301 

.98404 

45 

lb 

823 

113 

.5209 

399 

44 

17 

852 

143 

.5118 

394 

43 

18 

880 

173 

.5026 

389 

42 

19 
20 

21 

909 

203 

•4936 

383 

41 
40 

39 

.17937 

.18233 

5-4845 

•98378 

966 

263 

•4755 

373 

22 

.17995 

293 

.4665 

368 

38 

23 

.18023 

323 

•4575 

362 

37 

24 

052 

353 

.4486 

357 

36 

25 

.18081 

.18384 

5.4397 

-98352 

35 

2b 

109 

414 

.4308 

347 

34 

27 

138 

444 

.4219 

341 

33 

28 

166 

474 

•4131 

336 

32 

29 

30 

31 

^95 

504 

.4043 

33^ 

31 
30 

29 

.18224 

•18534 

5.3955 

•98325 

252 

564 

.3868 

320 

32 

281 

594 

•3781 

315 

28 

33 

309 

624 

•3694 

310 

27 

34 

338 

6S4 

.3607 

304 

26 

35 

.18367 

.18684 

5^3521 

.98299 

25 

3^^ 

395 

7'4 

•3435 

294 

24 

37 

424 

745 

•3349 

288 

23 

35^ 

452 

775 

.3263 

283 

22 

39 
40 

41 

481 

805 

.3178 

277 

21 
20 

19 

.18509 

.18835 

5.3093  1 .98272 

538 

865 

.3008 

267 

42 

567 

895 

.2924 

261 

18 

43 

595 

925 

.2839 

25b 

17 

44 

624 

955 

•2755 

250 

16 

45 

.18652 

.18986 

5.2672 

.98245 

IS 

'40 

681 

.19016 

.2588 

240 

14 

i  47 

710 

046 

.2505 

234 

13 

-  48 

738 

076 

.2422 

229 

12 

^9 

767 

106 

•2339 

223 

II 
10 

9 

50 

•18795 

.19136 

5-2257 

.98218 

^I 

824 

166 

.2174 

212 

,  ,2 

^5^ 

197 

.2092 

207 

8 

63 

881 

227 

.20II 

201 

7 

54 

910 

257 

.1929 

196 

6 

55 

.18938 

.19287 

5.1848 

.98190 

5 

5^ 

967 

.317 

•1.767 

185 

4 

57 

.18995 

347 

.1686 

179 

3 

158 

.19024 

378 

.1606 

174 

2 

052 

408 

.1526 

168 

0 

^0 

.19081 

.1943SI 

5.1446 

.98163 

;.cos. 

N^^CotJ 

IH.Tan.'N.Sin.l  '  | 

ir 

77 

/ 

N.Sin. 

N.Tan.N.Cot.|N.Cos 

0 

I 

.19081 

.19438 

5.1446 

.98163 

60 

S9 

109 

468 

.1366 

157 

2 

138 

498 

.1286 

152 

58 

3 

167 

529 

.1207 

146 

57 

4 

195 

559 

.1128 

140 

S6 

5 

.19224 

.19589 

5.1049 

•98135 

55 

6 

252 

619 

.0970 

129 

54 

7 

281 

649 

.0892 

124 

53 

8 

309 

680 

.0814 

118 

52 

9 
10 

II 

338 

710 

•0736 

112 

51 
50 

49 

.19366 

.19740 

5.0658 

.9«io7 

395 

770 

•0581 

lOI 

12 

423 

801 

•0504 

096 

48 

13 

452 

831 

.0427 

090 

47 

H 

481 

861 

•0350 

084 

46 

15 

.19509 

.19891 

50273 

.98079 

45 

16 

538 

921 

.0197 

073 

44 

17 

566 

952 

.0121 

067 

43 

18 

595 

.19982 

5.0045 

o6i 

42 

19 
20 

21 

623 

.20012 

4.9969 

056 

41 
40 

39 

.19652 

.20042 

4.9894 

.98050 

680 

073 

•9819 

044 

22 

709 

103 

•9744 

039 

38 

23 

737 

^33 

.9669 

033 

37 

24 

766 

164 

.9594 

027 

36 

25 

.19794 

.20194 

4.9520 

.98021 

35 

26 

823 

224 

.9446 

016 

34 

27 

851 

254 

•9372 

010 

33 

28 

880 

285 

.9298 

.98004 

32 

29 
30 

31 

908 

315 
•20345 

•9225 

.97998 

31 
30 

29 

•19937 

4.9152 

.97992 

965 

376 

.9078 

987 

32 

.19994 

406 

.9006 

981 

28 

33 

.20022 

436 

•8933 

975 

27 

34 

051 

466 

.8860 

969 

26 

35 

.20079 

.20497 

4.8788 

•97963 

25 

36 

108 

527 

.8716 

958 

24 

37 

136 

557 

.8644 

952 

23 

38 

165 

588 

•8573 

946 

22 

39 
40 

41 

193 

618 

•8501 

940 

21 
20 

19 

.20222 

.20648 

4.8430 

.97934 

250 

679 

•8359 

928 

42 

279 

709 

.8288 

922 

18 

43 

307 

739 

.8218 

916 

17 

44 

336 

770 

.8147 

910 

16 

45 

.20364 

.20800 

4.8077 

•97905 

15 

46 

393 

830 

.8007 

899 

14 

47 

421 

861 

•7937 

893 

13 

48 

450 

891   .7867 

887 

12 

49 
50 

51 

478 

921  1  -7798 

881 

II 
10 

9 

•20507 

.20952 

4.7729 

•97875 
869 

535 

.20982 

.7659 

52 

563 

.21013 

•7591 

863 

8 

53 

592 

043 

.7522 

857 

7 

54 

620 

073 

•7453 

8s. 

6 

55 

.20649 

.21104 

4.7385 

•97845 

S 

56 

677 

134 

■73^ 

839 

4 

57 

706 

164 

.7249 

833 

3 

58 

734 

195 

.7181 

2^7 

2 

59 
60 

763 

225 

.7114 

821 

0 

.20791 

.21256 

4.7046 

.97815 

N.Cos.l 

N.Cot.lN.Tan.,  N.Sin. 1 

/ 

7Q^. 


7QO 


78 

1 

2° 

13° 

1 

N.Sin. 

N.Tan. 

N.  Cot. 

N.Cos. 

1 

r 

N.Sin. 

N.Tan. 

N.  Cot. 

N.Cos. 

0 

.20791 

.21256 

4.7046 

.978i5 

60 

59 

0 

I 

.22495 

.23087 

4.3311 

•97437 

60 

59 

820 

286 

.6979 

809 

523 

117 

.3257 

430 

2 

848 

3it) 

.6912 

803 

58 

2 

552 

148 

.3200 

424 

58 

3 

«77 

347 

.6845 

797 

57 

3 

580 

179 

'3^43 

417 

57 

4 

905 

377 

.6779 

791 

56 

4 

608 

209 

.3086 

411 

56 

S 

.20933 

.21408 

4.6712 

.97784 

55 

5 

.22637 

.23240 

4.3029 

.97404 

55 

0 

962 

438 

.6646 

778 

54 

6 

665 

271 

.2972 

398 

54 

7 

.20990 

469 

.6580 

772 

53 

7 

693 

301 

.2916 

391 

53 

8 

.21019 

499 

.6514 

766 

S2 

8 

722 

332 

.2859 

384 

52 

9 
10 

11 

047 

529 

.6448 

760 

5i 
50 

49 

9 
10 

II 

750 

363 

.2803 

378 

51 
50  [ 

49 

.21076 

.21560 

4.6382 

.97754 

.22778 
807 

.23393 

4.2747 

•97371 

104 

590 

.6317 

748 

424 

.2691 

365 

12 

1.32 

621 

.6252 

742 

48 

12 

835 

455 

.2635 

358 

48 

13 

161 

651 

.6187 

735 

47 

13 

863 

485 

.2580 

351 

47 

'4 

189 

682 

.6122 

729 

46 

14 

892 

516 

.2524 

345 

46 

IS 

.21218 

.21712 

4.6057 

•97723 

45 

15 

.22920 

•23547 

4.2468 

.97338 

45 

i6 

246 

743 

.5993 

717 

44 

16 

948 

578 

.2413 

331 

44 

17 

275 

773 

.5928 

711 

43 

17 

.22977 

608 

.2358 

325 

43 

18 

303 

804 

.5864 

705 

42 

18 

.23005 

639 

•2303 

318 

42 

19 
20 

21 

331 

834 

.5800 

698 

41 
40 

39 

19 
20 

21 

033 

670 

.2248 

311 

41 
40 

39 

.21360 

.21864 

4.5736 

.97692 

.23062 

.23700 

4.2193 

.97304 

388 

89^ 

•5673 

686 

090 

731 

.2.39 

298 

22 

417 

925 

.5609 

680 

38 

22 

118 

762 

.2084 

291 

S8 

23 

445 

956 

•5546 

673 

37 

23 

146 

793 

.2030 

284 

37 

24 

474 

.21986 

•5483 

667 

36 

24 

175 

?'3 

.1976 

278 

36 

2S 

.21502 

.22017 

4.5420 

.97661 

35 

25 

.23203 

.23854 

4.1922 

.97271 

35 

26 

530 

047 

•5357 

655 

34 

26 

231 

885 

.1868 

264 

34 

27 

559 

078 

•5294 

648 

33 

27 

260 

916 

.1814 

257 

33 

28 

587 

108 

.5232 

642 

32 

28 

288 

946 

-1 760 

251 

32 

29 

30 

31 

616 

139 

.5169 

636 

31 
30 

29 

29 
30 

31 

316 

•23977 

.1706 

244 

31 
30 

29 

.21644 

.22169 

4.5107 

.97630 

•23345 

.24008 

4-1653 

•97237 

672 

200 

.5045 

623 

373 

039 

.1600 

230 

32 

701 

231 

.4983 

617 

28 

32 

401 

069 

•1547 

223 

28 

33 

729 

261 

.4922 

611 

27 

33 

429 

100 

•1493 

217 

27 

S4 

7S8 

292 

.4860 

604 

26 

34 

458 

131 

.1441 

210 

26 

3S 

.21786 

.22322 

4-4799 

.97598 

25 

35 

•23486 

.24162 

4.1388 

•97203 

25 

3^ 

814 

353 

•4737 

592 

24 

36 

5H 

193 

.1335 

196 

24 

37 

843 

383 

.4676 

585 

23 

37 

542 

223 

.1282 

189 

23 

3« 

871 

414 

.4615 

579 

22 

38 

571 

254 

.1230 

182 

22 

39 
40 

41 

899 

444 

•4555 

573 

21 
20 

19 

39 
40 

41 

599 

285 

.1178 

176 

21 
20 

19 

.21928 

.22475 

4.4494 

.97566 

.23627 

.24316 

4.1126 

.97169 

956 

505 

.4434 

560 

656 

347 

.1074 

162 

42 

.21985 

53^ 

•4373 

553 

18 

42 

684 

377 

.1022 

155 

18 

43 

.22013 

567 

•4313 

547 

17 

43 

712 

408 

.0970 

148 

17 

44 

041 

597 

•4253 

541 

16 

44 

740 

439 

.0918 

141 

16 

4S 

.22070 

.22628 

4.4194 

.97534 

15 

45 

.23769 

.24470 

4.0867 

•97134 

15 

46 

098 

658 

•4134 

528 

14 

46 

797 

501 

.0815 

127 

14 

47 

126 

689 

•4075 

521 

13 

47 

825 

532 

.0764 

120 

13 

48 

1^5 

719 

.4015 

51S 

12 

48 

853 

562 

.0713 

"3 

12 

49 
50 
51 

183 

750 

•3956 

508 

II 
10 

9 

49 
50 

51 

882 

593 

.0662 

.  106 

II 
10 

9 

.22212 

.22781 

4-3897 

.97502 

.23910 

.24624 

4.061 1 

.97100 

240 

811 

•3838 

496 

938 

6,^5 

.0560 

""^oi 

S2 

268 

842 

.3779 

489 

8 

52 

966 

686 

.0509 

086 

8 

53 

297 

872 

•3721 

483 

7 

53 

.23995 

717 

.0459 

079 

7 

S4 

325 

903 

.3662 

476 

6 

54 

.24023 

747 

.0408 

072 

6 

SS 

•22353 

•22934 

4.3604 

.97470 

5 

55 

.2405 1 

.24778 

40358 

.97065 

5 

56 

382 

964 

•3546 

463 

4 

56 

079 

809 

.0308 

058 

4 

S7 

410 

•22995 

.3488 

457 

3 

57 

108 

840 

.0257 

051 

3 

S8 

438 

.23026 

.3430 

450 

2 

58 

136 

871 

.0207 

044 

2 

59 
80 

467 

056 

•3372 

444 

I 
0 

59 
i60 

164 

902 

.0158 

037 

I 

22495 

.23087 

4-33'5 

•97437 

.24192 

•2A933 

4.0108 

.97030 

°i 

N.Cos. 

N.  Cot.  N.Tan. 

N.  Sin.|  ' 

L 

N.Cos. 

N.Cot.j  N.Tan. 

|N.Sin.|j^|| 

\ririo 


*ya' 


1 

4° 

1 

6° 

79 

/ 

N.Sin. 

N.Tan.iN.Cot. 

N.Cos.l   1 

t 

N.  Sin.  N. Tan,  N.  Cot.  N.Cos. 

0 

.24192 

.24933 

4.0108 

.97030 

60 

59 

0 

I 

.25882 

.20795 

3.7321 

•96593 

60 

S9 

220 

964 

.0058 

023 

910 

826 

.7277 

585 

2 

249 

.24995 

4.0009 

015 

58 

2 

938 

III 

.7234 

578 

S8 

3 

277 

.25020 

3-9959 

008 

57 

3 

966 

888 

.7191 

570 

57 

4 

305 

056 

.9910 

.97001 

S6 

4 

•25994 

920 

.7148 

562 

S6 

5 

.24333 

.25087 

3.9861 

.96994 

55 

5 

.26022 

.26951 

3.7105 

-96555 

SS 

6 

362 

118 

.9812 

987 

54 

6 

050 

.26982 

.7062 

547 

54 

7 

390 

149 

.9763 

980 

53 

7 

079 

.27013 

.7019 

540 

S3 

8 

418 

180 

.9714 

973 

S2 

8 

107 

044 

.6976 

532 

S2 

9 
10 

II 

446 

211 

.9665 

966 

51 

50 

49 

9 
10 

II 

135 

076 

•6933 

524 

5' 
50 

49 

.24474 

.25242 

3.9617 

.96959 

.26163 

.27107 

3.6891 

.96517 

503 

273 

.9568 

952 

191 

138 

.6848 

509 

12 

531 

304 

.9520 

945 

48 

12 

219 

169 

.6806 

502 

48 

13 

559 

33^ 

.9471 

937 

47 

13 

247 

201 

.6764 

494 

47 

'4 

S87 

366 

•9423 

930 

46 

H 

275 

232 

.6722 

486 

46 

IS 

.24615 

.25397 

3-9375 

•96923 

4S 

15 

.26303 

.27263 

3.6680 

.96479 

4S 

i6 

644 

.   428 

•9327 

916 

44 

16 

33^ 

294 

.6638 

471 

44 

17 

672 

459 

•9279 

909 

43 

17 

359 

326 

•6596 

463 

43 

18 

700 

490 

.9232 

902 

42 

18 

387 

357 

.6554 

456 

42 

19 
20 

21 

728 

521 

.9184 

894 

41 
40 

39 

19 
20 

21 

415 

388 

.6512 

448 

41 
40 

39 

.24756 

•25552 

3.9136 

.96887 

•26443 

.27419 

3.6470 

.96440 

784 

583 

.9089 

880 

471 

451 

.6429 

433 

22. 

813 

614 

.9042 

873 

38 

22 

500 

482 

.0387 

425 

38 

23 

841 

645 

.8995 

866 

37 

23 

528 

513 

•6346 

417 

37 

24 

869 

676 

.8947 

858 

36 

24 

556 

545 

•6305 

410 

36 

2S 

•24897 

.25707 

3.8900 

.96851 

3S 

2S 

.26584 

.27570 

3.6264 

.96402 

3S 

26 

925 

738 

.8854 

844 

34 

26 

612 

607 

.6222 

394 

34 

27 

954 

769 

.8807 

837 

33 

27 

640 

638 

.6181 

386 

33 

28 

.24982 

800 

.8760 

829 

32 

28 

668 

670 

.6140 

379 

32 

29 

30 

31 

.25010 

831 

.8714 

822 

31 
30 

29 

29 

30 

31 

.  696 

701 

.6100 

371 

31 
30 

29 

•25038 

.25862 

3-8667 

•96815 

.26724 

.27732 

36059 

.96363 

066 

893 

.8621 

807 

752 

764 

.6018 

355 

S2 

094 

924 

.8575 

800 

28 

32 

780 

795 

•5978 

347 

28 

33 

122 

955 

.8528 

793 

27 

33 

808 

826 

•5937 

340 

27 

34 

151 

.25986 

.8482 

786 

26 

34 

836 

858 

.5897 

332 

26 

3S 

■25179 

.26017 

3.8436 

•96778 

2S 

3S 

.26864 

.27889 

3.585^ 

.96324 

2S 

36 

207 

048 

.8391 

771 

24 

36 

892 

921 

.5816 

316 

24 

37 

235 

079 

.834? 

764 

23 

37 

920 

952 

•5776 

308 

23 

38 

263 

no 

.8299 

7S6 

22 

38 

948 

-27983 

.5736 

301 

22 

39 
40 

41 

291 

141 

.8254 

749 

21 

20 

19 

39 
40 

41 

.26976 

.28015 

.5696 

293 

21 

20 

19 

.25320 

.26172 

3.8208 

.96742 

.27004 

.28046 

35656 

.96285 

348 

203 

.8163 

734 

032 

077 

.5616 

277 

42 

376 

235 

.8118 

727 

18 

42 

060 

109 

•5576 

269 

18 

43 

404 

266 

.8073 

719 

17 

43 

088 

140 

•5536 

261 

'7 

44 

432 

297 

.8028 

712 

16 

44 

116 

172 

.5497 

253 

16 

4S 

.25460 

.26328 

3.7983 

.96705 

15 

4S 

.27144 

.28203 

3-5457 

.96246 

IS 

46 

488 

359 

.7938 

697 

14 

46 

172 

234 

.5418 

238 

14 

47 

S16 

390 

.7893 

690 

13 

47 

200 

266 

.5379 

230 

13 

48 

54^ 

421 

.7848 

682 

12 

48 

228 

297 

.5339 

222 

12 

49 
50 

SI 

573 

452 

.7804 

675 

II 
10 

9 

49 
50 

SI 

256 

329 

.5300 

214 

10 

.25601 

.26483 

3.7760 

.96667 

.27284 

.28360 

3.5261 

.96206 

629 

515 

•7715 

660 

312 

391 

.5222 

198 

S2 

6S7 

S4^ 

.7671 

6S3 

8 

S2 

340 

423 

-5183 

190 

8 

53 

685 

577 

.7627 

645 

7 

53 

368 

454 

•5 '44 

182 

7 

S4 

713 

608 

•7583 

638 

6 

S4 

396 

486 

•5105 

174 

6 

ss 

.25741 

.26639 

3-7539 

.96630 

5 

SS 

.27424 

.28517 

35067 

.96166 

S 

i6 

769 

670 

-7493 

623 

4 

56 

452 

549 

.5028 

158 

4 

S7 

798 

701 

.745' 

615 

3 

S7 

480 

580 

.4989 

150 

3 

sB 

826 

733 

.7408 

608 

2 

S8 

508 

612 

.4951 

142 

2 

59 
60 

854 

764 
.26795 

.7364 

600 

I 
0 

59 
60 

536 

643 

.4912 

134 

0 

.25882 

3-7321 

•96593 

.27564 

.28675 

3-4874 

.96126 

1 

N.Cos. 

N.Cot.lN.Tan. 

N.Sin. 

t 

N.Cos.  N.  Cot.  N. Tan. 

N.Sin. 

/ 

KJ^KO 


•7/1° 


8o 


16' 


ir 


N.  Sin, 


O 

I 

2 

3 

4 

5 
6 

7 
8 

9 

10 

II 

12 

13 
14 

15 
16 

17 

18 

19 
20 

21 
22 
23 
24 

25 
26 

27 
28 
29 

30 

31 
32 
33 
34 
35 
36 

37 
38 
39 
40 

41 
42 

43 
44 
45 
46 

47 
48 

49 

50 

51 

52 
53 
54 
55 
56 

57 
58 
59 
60 


27564 


592 
620 
648 
676 
27704 
731 
759 
787 
815 


•27843 


871 
899 
927 
95? 
•27983 
.28011 

039 
067 

095 


28123 


150 
178 
206 

234 

28262 
290 
318 
346 
374 


,28680 


708 
736 
764 

792 
.28820 
847 
875 
903 
931 


.28959 


,28987 

,29015 

042 

070 

,29098 

126 

209 


.29237 


N.Tan.  N.Cot. 


.28675 


706 
738 
769 

801 

,28832 

864 

895 
927 

958 


,28990 


29021 

053 

084 

116 
29147 
179 
210 
242 
274 


,29305 


Z?>7 
368 
400 

432 
,29463 

495 
526 
558 
590 


.29621 

"6^3" 
685 
716 

748 

.29780 

811 

843 
875 
906 


>38 


.2997Q 
.30001 

065 
.30097 
128 
160 
192 
224 


•30255 


287 
319 
351 
382 
.30414 
446 

478 
509 
541 


34874 


.4836 
•4798 
.4760 

.4722 
3.4684 
.4646 
.4608 
•4570 
•4533 


3^4495 


•4458 
.4420 

.4383 

.4346 

3-4308 

.4271 

•4234 
.4197 
.4160 


34124 


.4087 
•4050 
.4014 

•3977 
3-3941 
•3904 
.3868 
•3832 
.3796 


3-3759 

•3723 
.3687 
.3652 
.3616 
3-3580 
.3544 
•3509 
•3473 
•3438 


33402 


•3367 
•3332 
•3297 
.3261 
3.3226 
•3191 

•3156 
.3122 

•3087 


3-305: 


■30573 


N.Cos.lN.Cot 


•3017 
.2983 
.2948 

.2914 

32879 
•2845 
.2811 
•2777 
•2743 


3.2709 


N.Cos.l 


.96126 


118 
no 
102 

094 

,96086 

078 

070 
062 

054 


,96046 


037 

029 
021 

013 

,96005 
•95997 

989 
981 
972 


.95964 


956 
948 
940 

931 

•95923 
9'5 

907 
898 
890 


.95882 


874 
865 

857 

849 
,95841 
•832 
824 
816 
807 


•95799 


791 
782 
774 
766 
•95757 
749 
740 
732 
724 


•95715 


707 
698 
690 
681 

•95673 
664 

656 
647 
639 


.95630 


60 

59 
58 
57 
56 
55 
54 

53 
52 
51 
50 

49 
48 
47 
46 
45 
44 

43 
42 
41 
40 

39 
38 
37 
36 
35 
34 

33 
32 
31 
30 

29 
28 
27 
26 
25 
24 

23 
22 
21 

20 

^9 
18 

t7 
16 
15 
14 

13 
12 
II 

10 

9 

8 

7 
6 
5 
4 

3 


N.Tan. I  N.  Sin.| 


/ 

N.Sin. 

N.Tan. |N.  Cot. 

N.Cos". 

0 

.29237 

•30573 

3.2709 

-95630 

60 

S9 

265 

605 

.2675 

622 

2 

293 

637 

.2641 

613 

58 

3 

321 

669 

.2607 

60? 

57 

4 

348 

700 

•2573 

596 

56 

5 

.29376 

•30732 

32539 

.95588 

5S 

6 

404 

764 

.2506 

579 

54 

7 

432 

796 

.2472 

■571 

S3 

8 

460 

828 

.2438 

562 

S2 

9 
10 

II 

487 

860 

.2405 

554 

51 
50 

49 

•29515 

.30891 

3237 « 

•95545 

543 

923 

.2338 

536 

12 

571 

^55 

■2305 

528 

48 

13 

599 

•30987 

.2272 

519 

47 

14 

^26 

.31019 

•2238 

511 

46 

15 

.29654 

.31051 

3.2205 

•95502 

4S 

16 

682 

083 

.2172 

493 

44 

17 

710 

115 

.2139 

485 

43 

•  18 

737 

147 

.2106 

476 

42 

19 
20 

21 

765 

178 

•2073 

467 

41 
40 

39 

•29793 

.31210 

3.2041 

•95459 

821 

242 

.2008 

450 

22 

849 

274 

•i'975 

441 

38 

23 

876 

306 

•1943 

433 

37 

24 

904 

338 

.1910 

424 

36 

25 

.29932 

•31370 

3^1878 

•95415 

35 

26 

960 

402 

.1845 

407 

34 

27 

.29987 

434 

.1813 

398 

33 

28 

•30015 

466 

.1780 

389 

32 

29 
30 

31 

043 

498 

.1748 

380 

31 
30 

29 

.30071 

•31530 

3.1716 

•95372 

098 

562 

.1684 

363 

32 

126 

594 

.1652 

354 

28 

2,2, 

154 

626 

.1620 

'   345 

27 

^A 

182 

658 

.1588 

337 

26 

35 

.30209 

.31690 

3-1556 

•95328 

25 

36 

237 

722 

.1524 

319 

24 

37 

265 

754 

.1492 

310 

23 

38 

292 

786 

.1460 

301 

22 

39 
40 

41 

320 

818 

.1429 

293 

21 
20 

19 

.30348 

.31850 

3.1397 

.95284 

376 

882 

.1366 

275 

42 

403 

914 

•1334 

266 

18 

43 

431 

946 

.1303 

257 

17 

44 

459 

.31978 

.1271 

248 

16 

4S 

.30486 

.32010 

3.1240 

.95240 

15 

46 

5H 

042 

.1209 

231 

14 

47 

542 

074 

.1178 

222 

13 

48 

570 

106 

.1146 

213 

12 

49 
50 

SI 

597 

139 

.1115 

204 

II 
10 

9 

.30625 

.32171 

3.1084 

-95,195 

653 

203 

•1053 

186 

52 

680 

23s 

.1022 

177 

8 

53 

708 

267 

.0991 

168 

7 

S4 

736 

299 

.0961 

159 

6 

S5 

•30763 

.32331 

3.0930 

-95150 

5 

56 

791 

363 

.0899 

142 

4 

S7 

819 

396 

.0868 

133 

3 

S8 

846 

428 

.0838 

124 

2 

59 
60 

874 

460 

.0807 

115 

I 
0 

.30902 

.32492 

3^0777 

.95106 

iN.Cos. 

N.Cot.jN.Tan.l  N.Sin. 

/ 

•ys" 


79.° 


18' 


/ 

N.  Sin. 

N.Tan.iN.Cot. 

N.Cos. 

0 

.30902 

.32492 

3.0777 

.95106 

eo 

I 

929 

524 

.074 '> 

097 

59 

2 

95/ 

5.S6 

.0716 

088 

58 

3 

.309«5 

588 

.0086 

079 

57 

4 

.31012 

621 

•0655 

070 

56 

S 

.31040 

•32653 

3.0625 

.95061 

55 

6 

068 

685 

•0595 

052 

54 

7 

095 

717 

.0565 

043 

53 

8 

123 

749 

•0535 

033 

52 

9 
10 

II 

151 

782 

.0505 

024 

51 

50 

49 

.31178 

.32814 

3.0475 

•95015 

206 

846 

.0445 

,95006 

12 

233 

878 

.0415 

•94997 

48 

13 

261 

911 

.0385 

988 

47 

'4 

289 

943 

.0356 

979 

46 

i.S 

■3^31^ 

•32975 

3.0326 

.94970 

45 

i6 

344 

•33007 

.0296 

961 

44 

17 

372 

040 

.0267 

952 

43 

i8 

399 

072 

.0237 

943 

42 

19 
20 

21 

427 

104 

.0208 

933 

41 
40 

^2 

.31454 

•331.S6 

3.0178 

.94924 
915 

482 

169 

.0149 

22 

510 

201 

.0120 

906 

3^ 

23 

537 

233 

.0090 

897 

37 

24 

56^ 

266 

.0061 

888 

36 

2S 

.31593 

•33298 

3.0032 

•94878 

35 

26 

620 

330 

30003 

869 

34 

27 

648 

363 

2.9974 

860 

33 

28 

675 

395 

.9945 

851 

32 

29 

30 

703 

427 

.9916 

842 

31 
30 

29 

.31730 

.33460 

2.9887 
.9858 

.94832 

7S8 

492 

823 

32 

786 

524 

.9829 

814 

28 

33 

813 

557 

.9800 

805 

27 

34 

841 

589 

.9772 

795 

26 

3S 

.31868 

•33621 

2.9743 

•94786 

25 

3" 

896 

654 

.9714 

777 

24 

37 

923 

686 

.9686 

768 

23 

3H 

951 

718 

.9657 

758 

22 

39 
40 

41* 

•31979 

751 

.9629 

749 

21 
20 

19 

.32006 

■337^3 

2.9600 

.94740 

034 

816 

•9572 

730 

42 

061 

848 

.9544 

721 

18 

43 

089 

881 

•9515 

712 

n 

44 

116 

913 

.9487 

702 

16 

4"; 

.32144 

•33945 

2.9459 

•94693 

15 

46 

171 

•33978 

.9431 

684 

14 

47 

199 

.34010 

.9403 

674 

13 

48 

227 

043 

.9375 

605- 

12 

49 
50 

SI 

254 

075 

.9347 

656 

II 
10 

9 

.32282 

.34108 

2.9319 

.94646 

309 

f^o 

.9291 

637 

S2 

337 

173 

•9263 

627 

8 

53 

3H 

205 

•9235 

618 

7 

54 

392 

238 

.9208 

609 

6 

55 

.32419 

.34270 

2.9180 

•94599 

5 

5^ 

447 

303 

.9152 

590 

4 

57 

474 

335 

.9125 

580 

3 

5« 

502 

368 

.9097 

571 

2 

59 
60 

529 

400 

.9070 

561 

f 
0 

•32557 

.34433 

2.9042 

.94552 

N.  Cos.! N.  Cot.  N. Tan. 

N.Sin. 

f 

71° 


82 

20° 

/ 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.Cos. 

^ 

O 

I 

.34202 

•36397 

2-7475 

.93969 

60 

59 

229 

430 

•7450 

959 

2 

257 

463 

•7425 

949 

58 

3 

284 

496 

.7400 

939 

57 

4 

3" 

529 

.7376 

929 

56 

5 

.34339 

.36562 

2.7351 

•93919 

SS 

6 

366 

595 

.7326 

909 

54 

7 

393 

628 

.7302 

899 

S3 

8 

421 

6bi 

.7277 

889 

52 

9 
10 

II 

448 

694 

.7253 

879 

51 
50 

49 

•34475 

.36727 

2.7228 

.93869 

503 

7bo 

.7204 

859 

12 

530 

793 

.7179 

849 

48 

13 

557 

82b 

.7155 

839 

47 

14 

584 

859 

.7130 

829 

46 

15 

.34612 

.36892 

2.710b 

.93819 

45 

lb 

639 

925 

.7082 

809 

44 

17 

666 

958 

•7058 

799 

43 

i8 

694 

•36991 

.7034 

789 

42 

19 
20 

21 

721 

.37024 

.7009 

779 

41 
40 

39 

.34748 

.37057 

2.b985 

•93769 

775 

090 

.b9bi 

759 

22 

803 

123 

.6937 

748 

38 

23 

830 

157 

.6913 

738 

37 

24 

8S7 

190 

.b889 

728 

36 

25 

.34884 

•37223 

2.b8b5 

•9.3718 

3S 

2b 

912 

25b 

.b84i 

708 

34 

27 

939 

289 

.b8i8 

b98 

33 

28 

9bb 

322 

•6794 

b88 

32 

29 

30 

31 

.34993 

355 

.b77o 

677 

31 
30 

29 

.35021 

•37388 

2.b74b 

.93667 

048 

422 

.6723 

657 

32 

075 

455 

.bb99 

647 

28 

33 

102 

488 

.6675 

637 

27 

34 

130 

521 

.6652 

b2b 

2b 

35 

•35157 

•37554 

2.bb28 

•93616 

25 

3b 

184 

588 

.bbo5 

bob 

24 

37 

211 

b2I 

•6581 

596 

23 

3« 

239 

654 

•6558 

585 

22 

39 
40 

41 

26b 

b87 

•6534 

575 

21 

20 

19 

.35293 

.37720 

2.651 1 

.93565 

320 

754 

.b488 

5.S5 

42 

347 

787 

.b4b4 

544 

18 

43 

375 

820 

.b44i 

534 

17 

44 

402 

853 

.b4i8 

524 

lb 

45 

.35429 

•37887 

2.6395 

•93514 

15 

4b 

456 

920 

.6371 

503 

14 

47 

484 

953 

.6348 

493 

13 

48 

5" 

.37986 

•6325 

483 

12 

49 
50 

51 

538 

.38020 

.b302 

472 

10 

9 

•35565 

•38053 

2.b279 

.93462 

592 

08b 

.b25b 

452 

52 

bi9 

120 

•6233 

441 

8 

53 

647 

153 

.b2IO 

431 

7 

54 

'  674 

1 8b 

.bi87 

420 

b 

55 

•35701 

.38220 

2.blb5 

.93410 

5 

5b 

728 

253 

.bi42 

400 

4 

57 

755 

28b 

.bii9 

389 

3 

5^ 

320 

.bo9b 

379 

2 

59 
60 

810 

353 

.bo74 

368 

I 
0 

•35837 

•3838b 

2.6051 

•93358 

N.Cos. 

N.Cot.lN.Tan. 

iN.Sin.j  '  || 

2r 


/ 

N.Sin. 

N.Tan.  N.  Cot.|N.Cos.! 

1 

0 

I 

•35837 

.38386 

2.6051 

•93358 

60 

59 

8b4 

420 

.b028 

348 

2 

891 

453 

.boob 

337 

58 

3 

918 

487 

.5983 

327 

57 

4 

945 

520 

•5961 

316 

56 

5 

•35973 

•38553 

2.5938 

.93306 

55 

b 

.3booo 

587 

.5916 

295 

54 

7 

027 

b20 

.5893 

28s 

53 

8 

054 

654 

.5871 

274 

52 

9 
10 

II 

08 1 

b87 

.•5848 

2b4 

51 
50 

49 

.3bio8 

•38721 

2.582b 

.93253 

135 

754 

.5804 

243 

12 

Ib2 

787 

.5782 

232 

48 

13 

190 

821 

•5759 

222 

47 

14 

217 

854 

•5737 

211 

46 

15 

.36244 

.38888 

2.5715 

.93201 

45 

lb 

271 

921 

•5693 

190 

44 

17 

298 

955 

•5671 

180 

43 

18 

325 

.38988 

.5649 

ib9 

42 

19 
20 

21 

352 

.39022 

.5627 

159 

41 
40 

39 

.36379 

•39055 

2.5605 

.93148 

40b 

089 

.5583 

^37 

22 

434 

122 

.5561 

127 

38 

23 

4bi 

15b 

.5539 

lib 

37 

24 

488 

190 

.5517 

lob 

36 

25 

•36515 

.39223 

2.5495 

.93095 

35 

2b 

542 

257 

.5473 

084 

34 

27 

569 

290 

.5452 

074 

33 

28 

596 

324 

•5430 

063 

32 

29 

30 

31 

623 

357 

.5408 

052 

31 
30 

29 

.36650 

•39391 

2.5386 

.93042 

677 

425 

.5365 

031 

32 

704 

458 

.5343 

020 

28 

33 

731 

-492 

•5322 

.93010 

27 

34 

758 

S2b 

.5300 

.92999 

2b 

35 

•36785 

.39559 

2.5279 

.92988 

25 

36 

812 

593 

.5257 

978 

24 

37 

839 

b2b 

.5236 

967 

23 

38 

8b7 

bbo 

.5214 

956 

22 

39 
40 

41 

894 

694 

•5193 

945 

21 

20 

19 

.36921 

•39727 

2.5172 

.92935 

948 

7bi 

•5150 

924 

42 

•36975 

79-5 

•5129 

913 

18 

43 

.37002 

829 

.5108 

902 

17 

44 

029 

8b2 

.508b 

892 

lb 

45 

•37056 

•39896 

2.5065 

.92881 

15 

46 

083 

930 

.5044 

870 

H 

47 

no 

963 

.5023 

859 

13 

48 

137 

•39997 

.5002 

849 

12 

49 
50 

51 

ib4 

.40031 

.4981 

838 

II 

9 

•37191 

.400b5 

2.4960 

.92827 

218 

098 

.4939 

816 

52 

245 

132 

.4918 

805 

8 

53 

272 

ibb 

.4897 

794 

7 

54 

299 

200 

.4876 

784 

b 

55 

•37326 

.40234 

2.4855 

•92773 

5 

56 

353 

2b7 

.4834 

762 

4 

57 

380 

301 

.4813 

751 

3 

58 

407 

335" 

.4792 

740 

2 

59 
60 

434 

369  i  .4772 

729 

I 
0 

•37461 

.404031  2.4751 

.92718 

N.Cos. 

N.  Cot.  N.Tan. 

N.Sin. 

t 

fiS' 


68' 


22° 

23° 

83 

/ 

N.Sin. 

N.fan^jlN^Cot. 

N.Cos. 

/ 

N.  Sin.  N. Tan. !N.  Cot. 

N.Cos. 

0 

.37461 

.4040312.4751 

.92718 

60 

59 

0 

I 

.39073 

.42447 
482 

2.3559 

.92050 

60 

S9 

488 

436 

•4730 

707 

100 

.3539 

039 

2 

5'5 

470 

.4709 

697 

58 

2 

127 

516 

.3520 

028 

58 

3 

542 

504 

.4689 

686 

57 

3 

153 

551 

.3501 

016 

57 

4 

569 

538 

.4668 

675 

56 

4 

180 

585 

.3483 

•92005 

S6 

S 

•37595 

.40572 

2.4648 

.92664 

55 

5 

.39207 

.42619 

2.3464 

.91994 

55 

6 

622 

606 

.4627 

653 

54 

6 

234 

654 

.3445 

982 

54 

7 

649 

640 

.4606 

642 

53 

7 

260 

688 

.3426 

971 

53 

8 

676 

674 

.4586 

631 

52 

8 

287 

722 

.3407 

959 

52 

9 
10 

II 

703 

707 

.4566 

620 

51 
50 

49 

9 
10 

II 

3H 

757 

.3388 

948 

51 
50 

49 

•37730 

.40741 

2.4545 

.92609 

.39341 

.4279^ 

2.3369 

.91936 

757 

775 

.4525 

598 

367 

826 

•3351 

925 

12 

784 

809 

.4504 

587 

48 

12 

394 

860 

•3332 

914 

48 

13 

811 

843 

.4484 

576 

47 

13 

421 

894 

•.3313 

902 

47 

14 

838 

877 

.4464 

S6S 

46 

14 

448 

929 

•3294 
2.3276 

891 

46 

15 

•37«65 

.40911 

2.4443 

.92554 

45 

15 

•39474 

•42963 

.91879 

45 

i6 

892 

945 

.4423 

543 

44 

16 

501 

.42998 

.3257 

868 

44 

17 

919 

.40979 

■4403 

532 

43 

17 

528 

•43032 

•3238 

856 

43 

18 

946 

.41013 

.4383 

521 

42 

18 

555 

067 

.3220 

845 

42 

19 
20 

21 

973 

047 

.4362 

510 

41 
40 

^2 

19 
20 

21 

581 

lOI 

.3201 

833 

41 
40 

39 

.37999 

.41081 

2.4342 

.92499 

■39608 

.43136 

2.3183 

.91822 

.38026 

115 

.4322 

488 

635 

170 

.3164 

810 

22 

053 

149 

.4302 

477 

38 

22 

661 

205 

.3146 

799 

38 

23 

080 

183 

.4282 

466 

37 

23 

688 

239 

•3127 

787 

37 

24 

o'°7 

217 

.4262 

455 

36 

24 

715 

^74 

.3109 

775 

36 

25 

■38134 

.41251 

2.4242 

■92444 

35 

25 

•39741 

■43308 

2.3090 

.91764 

35 

26 

161 

285 

.4222 

432 

34 

26 

768 

343 

.3072 

752 

34 

27 

188 

319 

.4202 

421 

33 

27 

795 

378 

•3053 

741 

33 

28 

215 

353 

.4182 

410 

32 

28 

822 

412 

.3035 

729 

32 

29 
30 

31 

241 

387 

.4162 

399 

31 

30 

29 

29 
30 

31 

848 

447 

•3017 

718 

31 
30 

29 

.38268 

.41421 

2.4142 

.92388 

•39875 

■43481 

2.2998 

.91706 

295 

455 

.4122 

377 

902 

516 

.2980 

694 

32 

322 

490 

.4102 

366 

28 

32 

928 

550 

.2962 

683 

28 

33 

349 

524 

•4083 

355 

27 

33 

955 

585 

.2944 

671 

27 

34 

376 

558 

.4063 

343 

26 

34 

.39982 

620 

.2925 

660 

26 

35 

•38403 

.41592 

2.4043 

■92332 

25 

35 

.40008 

•43654 

2.2907 

.91648 

25 

36 

430 

626 

.4023 

321 

24 

36 

035 

689 

.2889 

636 

24 

37 

4S6 

660 

.4004 

310 

23 

37 

062 

724 

.2871 

625 

23 

^« 

483 

694 

.3984 

299 

22 

38 

088 

758 

.2853 

613 

22 

39 
40 

41 

510 

728 

.3964 

287 

21 
20 

19 

39 
40 

41 

"5 

793 

.2835 

601 

21 
20 

19 

.38537 

■41763 

2.394? 

.92276 

.40141 

.43828 

2.2817 

.91590 

564 

797 

.3925 

26s 

168 

862 

.2799 

578 

42 

591 

831 

.3906 

254 

18 

4^ 

195 

897 

.2781 

566 

18 

43 

617 

865 

.3886 

243 

17 

43 

221 

932 

.2763 

555 

17 

44 

644 

899 

.3867 

231 

16 

44 

248 

.43966 

.2745 

543 

16 

45 

.38671 

•41933 

2.3847 

.92220 

15 

45 

•40275 

.44001 

2.2727 

.91531 

15 

46 

698 

.41968 

.3828 

209 

14 

46 

301 

036 

.2709 

519 

14 

47 

725 

.42002 

.3808 

198 

13 

47 

328 

071 

.2691 

508 

13 

48 

752 

036 

.3789 

186 

12 

48 

355 

105 

.2673 

496 

12 

49 
50 

51 

778 

070 

•3770 

175 

II 
10 

9 

49 
50 

51 

381 

140 

.2655 

484 

II 
10 

9 

.38805 

.42105 

2-3750 

.92164 

.40408 

.44175 

2.2637 
.2620 

.91472 
461 

832 

139 

•3731 

152 

434 

210 

52 

8S9 

173 

•3712 

141 

8 

S2 

461 

244 

.2602 

T^ 

8 

53 

886 

207 

•3693 

130 

7 

53 

488 

279 

.2584 

7 

54 

912 

242 

•3673 

119 

6 

54 

514 

3H 

.2566 

425 

6 

55 

■38939 

.42276 

2.3654 

.92107 

5 

55 

.40541 

.44349 

2.2549 

.91414 

5 

S^ 

966 

310 

.3635 

096 

4 

56 

567 

384 

.2531 

402 

4 

57 

■38993 

345 

.3616 

085 

3 

57 

594 

418 

.2513 

390 

3 

5« 

.39020 

379 

.3597 

073 

2 

58 

621 

453 

.2496 

378 

2 

59 
60 

046 

413 

.3578 

062 

L 
0 

59 
60 

647 

488 

.2478 

366 

0 

•39073 

•42447 !  2.3559 

.92050 

.40674 

■44523 

2.2460 

.91355 

N.Cos. 

N.  Cot.  N. Tan. 

N.Sin.l  '   \ 

__ 

N.Cos.iN.Cot.  N.Tan. 

N.Sin. 

/ 

fir 


fifi° 


84 

24° 

1 

N.  Sin.|N.Tan.iN.Cot.|N.Cos.|   || 

0 

I 

.40674 

•44523 

2.2460 

•91355 

60 

59 

700 

558 

•2443 

343 

2 

727 

593 

•2425 

2)2>^ 

58 

3 

753 

627 

.2408 

319 

57 

4 

780 

662 

.2390 

307 

56 

5 

.40806 

.44697 

2.2373 

.91295 

55 

b 

^Z?> 

732 

•2355 

283 

54 

,7 

860 

767 

.2338 

272 

S3 

8 

886 

802 

.2320 

260 

52 

9 
10 

913 

837 

•2303 

248 

51 
50 

49 

.40939 

.44872 

2.228b 

.91236 

966 

907 

.2268 

224 

12 

.40992 

942 

.2251 

212 

48 

13 

.41019 

•44977 

.2234 

200 

47 

14 

045 

.45012 

.2216 

188 

46 

15 

.41072 

•45047 

2.2199 

.91176 

45 

lb 

098 

082 

.2182 

164 

44 

'Z 

125 

117 

.2165 

152 

43 

18 

151 

152 

.2148 

140 

42 

19 
20 

21 

178 

187 

.2130 

128 

41 
40 

39 

.41204 

.45222 

2.2113 

.91116 

231 

257 

.2096 

104 

22 

257 

292 

.2079 

092 

38 

23 

284 

327 

.2062 

080 

37 

24 

310 

362 

.2045 

068 

36 

25 

•41337 

•45397 

2.2028 

.91056 

35 

2b 

Z^Z 

432 

.2011 

044 

34 

^Z 

390 

467 

.1994 

032 

ZZ 

28 

416 

502 

.1977 

020* 

32 

29 
30 

31 

443 

53« 

.i960 

.91008 

31 
30 

29 

.41469 

•45573 

2.1943 

.90996 

496 

608 

.1926 

984 

32 

522 

643 

.1909 

972 

28 

33 

549 

678 

.1892 

960 

27 

34 

575 

l^Z 

.1876 

948 

26 

35 

.41602 

•45748 

2.1859 

.90936 

25 

3^ 

628 

784 

.1842 

924 

24 

37. 

65s 

819 

.1825 

911 

23 

3« 

68i 

854 

.1808 

899 

22 

39 
40 

41 

707 

889 

.1792 

887 

21 
20 

19 

•41734 

.45924 

2-1775 

.90875 

760 

960 

.  -1758 

863 

42 

787 

•45995 

.1742 

851 

18 

43 

813 

.46030 

.1725 

839 

17 

44 

840 

065 

.1708 

826 

16 

45 

.41866 

.46101 

2.-1692 

.90814 

IS 

46 

892 

136 

•1675 

802 

H 

47 

919 

171 

.1659 

790 

13 

48 

945 

206 

.1642 

778 

12 

49 
50 

51 

972 

242 

.1625 

766 

11 
10 

9 

.41998 

.46277 

2.1609 

•90753 

.42024 

312 

•1592 

741 

52 

051 

348 

•1576 

729 

8 

53 

077 

383 

.1560 

7'7 

7 

54 

104 

418 

•1543 

704 

6 

55 

.42130 

.46454 

2.1527 

.90692 

5 

5<^ 

156 

489 

.1510 

680 

4 

57 

183 

525 

.1494 

668 

3 

5^ 

209 

560 

.1478 

6SS 

2 

59 
60 

235 

595  !  -^61  i 

643 

I 

0 

.42262  1  .46631  1  2.1445  j  .90631 1 

N.Cos.|n.  Cot.|N.Tan.!  N.  Sin.| 

^J 

26' 


f 

N.  Sin.  N.Tan.  N.  Cot.|N.Cos. 

0 

.42262 

.46631 

2.1445 

•90631 

60 

S9 

288 

666 

.1429 

618 

2 

315 

702 

•1413 

606 

58 

3 

341 

737 

.1396 

594 

57 

4 

367 

772 

.1380 

S82 

S6 

5 

.42394 

.46808 

2.1364 

.90569 

SS 

b 

420 

843 

.1348 

557 

54 

7 

446 

879 

•1332 

545 

53 

8 

473 

914 

•i3'5 

532 

52 

9 
10 

II 

499 

950 

.1299 

520 

51 
50 

49 

•42525 

.46985 

2.1283 

.90507 

552 

.47021 

.1267 

495 

12 

578 

056 

•1251 

483 

48 

13 

604 

092 

•1235 

470 

47 

14 

631 

128 

.1219 

458 

46 

15 

.42657 

•47163 

2.1203 

.90446 

4S 

lb 

683 

199 

.1187 

433 

44 

17 

709 

234 

.1171 

421 

43 

,18 

73t> 

270 

.1155 

408 

42 

19 
20 

21 

762 

305 

•1139 

396 

41 
40 

39 

.42788 

•47341 

2.1123 

.90383 

8'5 

377 

.1107 

371 

22 

841 

412 

.1092 

3S8 

38 

23 

867 

448 

.1076 

346 

37 

24 

894 

483 

.1060 

334 

36 

2S 

.42920 

•47519 

2.1044 

.90321 

3S 

26 

946 

555 

.1028 

309 

34 

27 

972 

590 

.1013 

296 

33 

28 

.42999 

626 

.0997 

284 

32 

29 

30 

31 

.43025 

662 

.0981 

271 

31 
30 

29 

•43051 

.47698 

2.0965 

.90259 

077 

IZZ 

.0950 

246 

32 

104 

769 

.0934 

233 

28 

zz 

130 

805 

.0918 

221 

27 

34 

IS6 

840 

.0903 

208 

26 

35 

.43182 

.47876 

2.0887 

.90196 

25 

3<^ 

209 

912 

.0872 

183 

24 

37 

2.SS 

948 

.0856 

171 

23 

38 

261 

•47984 

.0840 

IS8 

39 
40 

41 

287 

.48019 

.0825 

146 

21 
20 

19 

.43313 

•48055 

2.0809 

•90133 

340 

091 

.0794 

120 

42 

366 

127 

.0778 

108 

18 

43 

392 

163 

.0763 

095 

17 

44 

418 

198 

.0748 

082 

16 

45 

•43445 

.48234 

2.0732 

.90070 

15 

46 

471 

270 

.0717 

057 

14 

47 

■  497 

306 

.0701 

045 

13 

48 

523 

342 

.0086 

032 

12 

49 
50 
SI 

549 

378 

.0671  i  ,  019 1 

II 
10 

9 

•43575 

.48414 

2.0655 

.90007 

602 

450 

.0640 

.89994 

S2 

628 

486 

.0625 

981 

8 

53 

654 

521 

.0609 

968 

7 

S4 

680 

557 

•0594 

956 

6 

ss 

.43706 

•48593 

2.0579 

•89943 

5 

56 

733 

629 

.0564 

930 

4 

57 

759 

665 

•0549 

918 

3 

58 

785 

701 

•0533 

905 

2 

59 
60 

811 

737 

.0518 

892 

I 
0 

•43837 

•48773 

2.0503 

.89879 

N.Cos.l 

N.  Cot.  N.Tan.  !N.Sin.| 

/ 

fi/i° 


64" 


26° 


'  iN.Sin.JN.Tan.N.Cot. 

N.Cos 

1 

0 

I 

•43837 

.48773 

2.0503 

.89879 

60 

59 

863 

809 

.0488 

867 

2 

.  889 

845 

•0473 

854 

S8 

3 

916 

881 

.0458 

841 

57 

4 

942 

917 

•0443 

828 

56 

S 

.43968 

•489^3 

2.0428 

.89816 

55 

6 

•43994 

.48989 

.0413 

803 

54 

7 

.44020 

.49026 

.0398 

790 

5^ 

8 

046 

062 

•0383 

777 

52 

9 
10 

II 

072 

098 

.0368 

764 

51 
50 

49 

.44098 

.49134 

2.0353 

.89752 

124 

170 

•0338 

739 

12 

151 

206 

•0323 

726 

48 

13 

177 

242 

.0308 

713 

47 

H 

203 

278 

.0293 

700 

46 

IS 

.44229 

•49315 

2.0278 

.89687 

45 

It) 

255 

351 

.0263 

674 

44 

17 

28^ 

387 

.0248 

662 

43 

18 

307 

423 

•0233 

649 

42 

19 
20 

21 

333 

459 

.0219 

636 

41 
40 

39 

•44359 

•49495 

2.0204 

.89623 

385 

532 

.0189 

610 

22 

411 

568 

.0174 

597 

38 

23 

437 

604 

.0160 

584 

37 

24 

464 

640 

.0145 

571 

36 

25 

.44490 

.49677 

2.0130 

•89558 

35 

26 

51b 

713 

.0115 

545 

34 

27 

542 

749 

.0101 

532 

33 

28 

568 

786 

.0086 

519 

32 

29 
30 

31 

594 

822 

.0072 

506 

3J 
30 

29 

.44620 

.49858 

2.0057 

.89493 

646 

894 

.0042 

480 

32 

672 

931 

.0028 

467 

28 

33 

698 

.49967 

2.0013 

454 

27 

34 

724 

.50004 

1.9999 

441 

26 

35 

•44750 

.50040 

1.9984 

.89428 

25 

3(v 

77b 

076 

.9970 

415 

24 

37 

802 

"3 

•9955 

402 

23 

38 

828 

149 

•9941 

389 

22 

39 
40 

41 

854 
.44880 

185 

.9926 

376 

21 
20 

19 

.50222 

1.9912 

.89363 

906 

258 

•9897 

350 

42 

932 

295 

.9883 

337 

18 

43 

958 

33i 

.9868 

324 

17 

44 

.44984 

368 

•9854 

3" 

16 

45 

.45010 

.50404 

1.9840 

.89298 

15 

46 

036 

441 

.9825 

285 

14 

47 

062 

477 

.9811 

27^ 

13 

48 

088 

5H 

.9797 

259 

12 

49 
59 

51 

114 

550 

.9782 

245 

II 
10 

9 

.45140 

•50587 

1.9768 

.89232 

166 

623 

•9754 

219 

52 

192 

660 

.9740 

206 

8 

53 

218 

696 

•9725 

193 

7 

54 

243 

733 

.9711 

180 

6 

55 

.45269 

.50769 

1.9697 

.89167 

S 

5^ 

295 

806 

.9683 

153 

4 

57 

321 

843 

.9669 

140 

3 

5^ 

347 

879 

.9654 

127 

2 

59 

373 

916 

.9640 

114 

I 

60 

•45399 

•50s  53 

1.9626 

.89101 

0 

|N.Cos.|N.Cot.  N.Tan.!N.Sin.| 

/ 

27° 

85 

'  (n-  Sin.  N.Tan.  N.  Cot.  N^ Cos. 

0 

I 

.45399 
425 

.50953 
.50989 

1.9626 
.9612 

.89101 

60 

59 

087 

2 

45' 

.51026 

.9598 

074 

58 

3 

477 

063 

■9584 

061 

57 

4 

503 

099 

•9570 

048 

56 

5 

.45529 

.51136 

1.9556 

.89035 

55 

6 

554 

173 

•9542 

021 

54 

7 

580 

209 

.9528 

.89008 

53 

8 

606 

246 

.95 '4 

•88995 

52 

9 
10 

II 

632 

283 

.9500 

981 

51 
50 

49 

•45658 

•51319 

1.9486 

.88968 

684 

356 

•9472 

955 

12 

710 

393 

•9458 

942 

48 

13 

736 

430 

■9444 

928 

47 

14 

762 

467 

.9430 

915 

46 

15. 

•45787 

•51503 

1.9416 

.88902 

45 

16 

813 

540 

.9402 

888 

44 

17 

839 

577 

.9388 

875 

43 

18 

865 

614 

.9375 

862 

42 

19 
20 

21 

891 

651 

.9361 

848 

41 
40 

39 

•45917 

.51688 

1.9347 

.88835 

942 

724 

.9333 

822 

22 

968 

761 

•9319 

808 

38 

23 

•45994 

798 

.9306 

795 

37 

24 

.46020 

835 

.9292 

782 

36 

25 

.46046 

.51872 

1.9278 

.88768 

35 

26 

072 

909 

.9265 

755 

34 

27 

097 

946 

.9251 

741 

33 

28 

123 

•51983 

•9237 

728 

32 

29 
30 

31 

149 

.52020 

.9223 

715 

31 
30 

29 

•46175 

•52057 

1.9210 

.88701 

201 

094 

.9196 

688 

32 

226 

131 

.9183 

674 

28 

33 

252 

168 

.9169 

661 

27 

34 

278 

205 

•9155 

647 

26 

35 

.46304 

.52242 

1.9142 

.88634 

25 

36 

330 

279 

.9128 

620 

24 

37 

355 

316 

•9115 

607 

23 

38 

381 

•  353 

.9101 

593 

22 

39 
40 

41 

407 

390 

.9088 

580 

21 
20 

19 

•46433 

•52427 

1.9074 

.88566 

458 

464. 

.9061 

553 

42 

484 

5°i 

.9047 

539 

18 

43 

510 

538 

•9034 

526 

17 

44 

536 

575 

.9020 

512 

16 

45 

.46561 

.52613 

1.9007 

.88499 

15 

46 

587 

650 

•8993 

485 

14 

47 

613 

687 

.8980 

472 

13 

48 

639 

724 

.8967 

458 

12 

49 
50 

51 

664 

761 

.8953 

445 

II 
10 

9 

.46690 

•52798 

1.8940 

.88431. 

716 

836 

.8927 

417 

52 

742 

873 

•8913 

404 

8 

53 

767 

910 

.8900 

390 

7 

54 

793 

947 

.8887 

377 

6 

55 

.46819 

•52985 

1.8873 

.88363 

5 

56 

844 

.53022 

.8860 

349 

4 

57 

870 

059 

.8847 

336 

3 

58 

896 

09l6 

.8834 

322 

2 

59 
60 

921 

134 

.8820 

308 

I 
0 

.46947 

•53171 

1.8807 

.88295 

N.Cos.lN.Cot.l 

N.Tan. 1  N.  Sin.| 

r 

fiOo 


on° 


86 

28° 

' 

N.  Sin. 

N.Tan.  N.Cot.iN.Cos.l   || 

O 

I 

.46947 

•53171 

1.8807 

.88295 

60 

59 

973 

208 

•8794 

281 

2 

.46999 

246 

.8781 

267 

58 

3 

.47024 

283 

.8768 

254 

57 

.  4 

050 

320 

•8755 

240 

S6 

5 

.47076 

•53358 

1.8741 

.88226 

ss 

to 

lOI 

395 

.8728 

213 

54 

7 

127 

432 

.8715 

199 

53 

8 

153 

470 

.8702 

i8s 

S2 

9 
10 

II 

178 

507 

.8689 

172 

51 
50 

49 

.47204 

•53545 

1.8676 

.88158 

229 

582 

.8663 

144 

12 

255 

620 

.8650 

130 

48 

13 

281 

657 

.8637 

117 

47 

H 

306 

694 

.8624 

103 

46 

15 

.4733^ 

•53732 

1.8611 

.88089 

45 

lb 

35« 

769 

.8598 

075 

44 

17 

383 

807 

.8585 

062 

43 

18 

409 

844 

.8572 

048 

42 

19 
20 

21 

434 

882 

•8559 

034 

41 
40 

39 

.47460 

.53920 

1.8546 

.88020 

486 

957 

•8533 

.88006 

22 

511 

•53995 

.8520 

•87993 

38 

^3 

537 

.54032 

.8507 

979 

37 

24 

562 

070 

.8495 

965 

36 

25 

■4758» 

•54107 

1.8482 

•87951 

35 

26 

614 

145 

.8469 

937 

34 

27 

639 

183 

.8456 

923 

33 

28 

66^ 

220 

•8443 

909 

32 

29 
30 

31 

690 

258 

.8430 

896 

31 
30 

29 

.47716 

•54296 

1.8418 

.87882 

741 

333 

.8405 

868 

32 

767 

371 

.8392 

8S4 

28 

33 

793 

409 

•8379 

840 

27 

34 

818 

446 

.8367 

826 

26 

35 

.47844 

•54484 

1^8354 

.87812 

25 

3^ 

869 

522 

.8341 

798 

24 

37 

895 

560 

.8329 

784 

23 

3° 

920 

597 

.8316 

770 

22 

39 
40 

41 

946 

635 

.8303 

756 

21 
20 

19 

.47971  .54673!  1-8291 

•87743 

•47997 

711 

.8278 

729 

42 

.48022 

748 

.8265 

71S 

18 

43 

048 

786 

.8253 

701 

17 

44 

073 

824 

.8240 

687 

16 

45 

.48099 

.54862 

1.8228 

•87673 

15 

4b 

124 

900 

.8215 

659 

14 

47 

150 

938 

.8202 

645 

13 

48 

175 

.54975 

.8190 

631 

12 

49 
50 

51 

201 

•55013  ^8177 

617 

II 
10 

9 

.48226 

.55051  I  1.8165 

.87603 

252 

089 

.8152 

589 

52 

277 

127 

.8140 

575 

8 

53 

303 

165 

.8127 

561 

7 

54 

328 

203 

•8115 

546 

6 

55 

•48354 

.55241 

1.8103 

•87532 

5 

5t> 

379 

279 

.8090 

518 

4 

57 

405 

3^7 

.8078 

504 

3 

5^^ 

430 

355  ^8065 

490 

2 

59 
60 

456 

393  1  .8053 

476 

0 

.48481 

.55431  !  1.8040 

.87462 

N.Cos.JN.Cot.lN.Tan. 

N.Sin.|  '  1 

29' 


/ 

N.Sin.lN.Tan. 

N.  Cot. 

N.Cos.l 

0 

I 

.48481 

.55431 

1.8040 

.87462 

60 

59 

506 

469 

.8028 

448 

2 

532 

507 

.8016 

434 

58 

3 

557 

545 

.8003 

420 

57 

4 

583 

583 

.7991 

406 

56 

5 

.48608 

•55621 

1.7979 

-87391 

55 

b 

634 

659 

.7966 

377 

54 

7 

659 

697 

.7954 

363 

53 

8 

684 

736 

.7942 

349 

52 

9 
fO 

II 

710 

774 

•7930 

335 

51 
50 

49 

.48735  i  .55812 

1.7917 

-87321 

761 

850 

•.79<55 

306 

12 

786 

888 

•7893 

292 

48 

13 

811 

926 

.7881 

278 

47 

14 

837 

•55964 

.7868 

264 

46 

15 

.48862 

•56003 

1.7856 

.87250 

45 

lb 

888 

041 

•7844 

235 

44 

17 

913 

079 

.7832 

221 

43 

18 

938 

117 

.7820 

207 

42 

19 
20 

21 

964 

15b 

.7808 

193 

41 
40 

39 

.48989 

•56194 

1.7796 

.87178 

.49014 

232 

.7783 

164 

22 

040 

270 

.7771 

150 

38 

23 

065 

309 

•7759 

136 

37 

24 

090 

347 

•7747 

121 

36 

25 

.49116 

•56385 

1-7735 

.87107 

35 

2b 

141 

424 

•7723 

093 

34 

27 

166 

462 

.7711 

079 

33 

28 

192 

501 

•7699 

064 

32 

29 

30 

217 

539 

•7687 

050 

31 
30 

29 

.49242 

•56577 

1.7675 

.87036 

268 

616 

.7663 

021 

32 

293 

654 

•7651 

.87007 

28 

33 

318    693 

•7639 

.86993 

27 

34 

344 

731 

•7627 

978 

26 

3S 

•49369 

.56769 

17615 

.86964 

25 

36 

394 

808 

•7603 

949 

24 

37 

419 

846 

•7591 

935 

23 

3ii 

445 

885 

•7579 

921 

22 

39 
40 

41 

470 

923 

.7567 

90b 

21 
20 

19 

•49495 

.56962 

1-7556 

.86892 

521 

.57000 

•7544 

878 

42 

546 

039 

•7532 

863 

18 

43 

571 

078 

.7520 

849 

17 

44 

596 

116 

.7508 

834 

16 

45 

.49622 

•57155 

1.7496 

.86820 

15 

46 

647 

193 

•7485 

805 

14 

47 

672 

232 

•7473 

791 

13 

48 

697 

271 

.7461 

777 

12 

49 
50 

51 

723 

309 

.7449 

7b2 

II 
10 

Q 

•49748 

.57348 

1-7437 
.7426 

.86748 

773 

386 

733 

52 

798 

425 

.7414 

719 

8 

53 

824 

464 

.7402 

704 

7 

S4 

849 

503 

•7391 

690 

6 

55 

.49874 

•57541 

1-7379 

.86675 

5 

56 

899 

580 

•7367 

661 

4 

57 

924 

619 

•7355 

646 

3 

58 

950 

657 

•7344 

632 

2 

59 
60 

•49971 

696 

-7332 

617 

0 

.50000 

•57735 

1.7321 

.86603 

N.Cos. 

N.Cot. 

N.Tan.!  N.  Sin. 

/ 

30° 

31° 

87 

1 

N.Sin. 

N.Tan^ 

N.  Cot. 

N.Cos. 

' 

N.Sin.  N.Tan.|N.  Cot. 

N.Cos. 

0 

.5CXXX3 

•57735 

1. 7321 

.86603 

60 

59 

0 

I 

.51504 

.60086 

1.6643 

.85717 

60 

59 

025 

774 

.7309 

'   588 

529 

126 

.6632 

702 

2 

050 

813 

.7297 

573 

58 

2 

554 

165 

.6621 

687 

58 

3 

076 

851 

.7286 

559 

57 

3 

579 

205 

.6610 

672 

57 

4 

lOI 

890 

•7274 

544 

56 

4 

604 

24^ 

.6599 

657 

56 

S 

.50126 

•57929 

1.7262 

.86530 

55 

5 

.51628 

.60284 

1.6588 

•85642 

55 

6 

151 

.57968 

•7251 

515 

54 

6 

653 

324 

.6577 

627 

54 

7 

176 

.58007 

•7239 

501 

53 

7 

678 

364 

.6566 

612 

53 

8 

201 

046 

.7228 

486 

52 

8 

703 

403 

•6555 

597 

52 

9 
10 

227 

085 

.7216 

471 

51 
50 

49 

9 
10 

II 

728 

443 

.6545 

582 

51 
50 

49 

.50252 

.58124 

1.7205 

.86457 

•51753 

.60483 

1.6534 

.85567 

277 

162 

.7193 

442 

778 

522 

.6523 

551 

12 

302 

201 

.7182 

427 

48 

12 

803 

562 

.6512 

536 

48 

13 

327 

240 

.7170 

413 

47 

13 

828 

602 

.6501 

521 

47 

H 

352 

279 

•7159 

398 

46 

, 

14 

852 

642 

.6490 

506 

46 

15 

•50377 

.58318 

1. 7147 

.86384 

45 

15 

.51877 

.60681 

1.6479 

-85491 

45 

16 

403 

357 

.713b 

369 

44 

16 

902 

721 

.6469 

476 

44 

17 

428 

396 

.7124 

354 

43 

17 

927 

761 

.6458 

461 

43 

18 

453 

435 

'I^^Z 

340 

42 

18 

952 

801 

.6447 

446 

42 

19 
20 

21 

478 

474 

.7102 

325 

41 
40 

39 

19 
20 

21 

•51977 

841 

.6436 

431 

41 
40 

39 

.50503 

■58513 

1.7090 

.86310 

.52002 

.60881 

1.6426 

•85416 

528 

552 

.7079 

295 

026 

921 

.6415 

401 

22 

553 

591 

.7067 

281 

38 

22 

051 

.60960 

.6404 

.385 

38 

23 

578 

631 

.7056 

266 

37 

23 

076 

.61000 

•6393 

370 

Z1 

24 

603 

670 

.7045 

251 

36 

24 

lOI 

040 

■6383 

355 

36 

2S 

.50628 

.58709 

1.7033 

.86237 

35 

25 

.52126 

.61080 

1.6372 

.85340 

35 

26 

654 

748 

.7022 

222 

34 

.26 

151 

120 

.6361 

325 

34 

27 

679 

787 

.7011 

207 

33 

27 

'75 

160 

.6351 

310 

ZZ 

28 

704 

826 

.6999 

192 

32 

28 

200 

200 

.6340 

294 

32 

29 

30 

31 

729 

865 

.6988 

178 

31 
30 

29 

29 
30 

SI 

225 

240 

.6329 

279 

31 
30 

29 

•50754 

.58905 

1.6977 

.86163 

•52250 

.61280 

1-6319 

.85264 

'779 

944 

•6965 

148 

275 

320 

.6308 

249 

32 

804 

.58983 

•6954 

nz 

28 

32 

299 

360 

.6297 

234 

28 

zz 

829 

.59022 

•6943 

119 

27 

zz 

324 

400 

.6287 

218 

27 

34 

854 

061 

.6932 

104 

26 

S4 

349 

440 

.6276 

203 

26 

35 

•50879 

.59101 

1.6920 

.86089 

25 

35 

.52374 

.61480 

1.6265 

.85188 

25 

3^ 

904 

140 

.6909 

074 

24 

36 

399 

520 

.6255 

173 

24 

37 

929 

179 

.6898 

059 

23 

37 

423 

561 

.6244 

157 

23 

3« 

954 

218 

.6887 

045 

22 

38 

448 

601 

.6234 

142 

22 

39 
40 

41 

•50979 

258 

.6875 

030 

21 
20 

19 

39 
40 

41 

473 

641 

.6223 

127 

21 
20 

19 

.51004 

•59297 

1.6864 

.86015 

.52498 

.61681 

1. 6212 

.85112 

029 

336 

.6853 

.86000 

522 

721 

.6202 

096 

42 

054 

376 

.6842 

.85985 

18 

42 

547 

761 

.6191 

081 

18 

43 

079 

415 

.6831 

970 

17 

43 

572 

801 

.6181 

066 

17 

44 

104 

454 

.6820 

956 

16 

44 

597 

842 

.6170 

051 

16 

4S 

.51129 

.59494 

1.6808 

•85941 

15 

45 

.52621 

.61882 

1. 6160 

•85035 

15 

46 

154 

533 

.6797 

'926 

14 

46 

646 

922 

.6149 

020 

14 

47 

179 

573 

.6786 

911 

13 

47 

671 

.61962 

.6139 

•85005 

13 

48 

204 

612 

.6775 

896 

12 

48 

696 

.62003 

.6128 

.84989 

12 

49 
50 

SI 

229 

651 

.6764 

881 

II 
10 

9 

49 
50 

51 

720 

043 

.6118 

974 

II 
10 

9 

•51254 

.59691 

1-6753 

.85866 

•52745 

.62083 

1. 6107 

■84959 

279 

730 

.6742 

851 

770 

124 

.6097 

943 

S2 

304 

770 

•6731 

836 

8 

52 

794 

164 

.6087 

928 

8 

53 

329 

809 

.6720 

821 

7 

53 

819 

204 

.6076 

913 

7 

54 

354 

849 

.6709 

806 

6 

54 

844 

245 

.6066 

897 

6 

55 

•51379 

•59888 

1.6698 

•85792 

5 

55 

.52869 

.62285 

1-6055 

.84882 

5 

5^ 

404 

928 

.6687 

777 

4 

56 

893 

325 

,6045 

866 

4 

57 

429 

•59967 

.6676 

762 

3 

57 

918 

366 

.6034 

851 

3 

58 

454 

.60007 

.6665 

747 

2 

58 

943 

406 

.6024 

836 

2 

59 
60 

479 

046 

.6654 

732 

0 

59 
60 

967 

446 

.6014 

820 

0 

■51504 

.60086 

1.6643 

.85717 

.52992 

.62487 

1.6003 

.84805 

N.Cos.!N.Cot.iN.Tan.|  N.Sin. 

/■ 

N.Cos.|N.Cot. 

N.Tan. 

N.Sin. 

t 

KQ^ 


Kft°' 


88 

32° 

t 

N.  Sin.|N.Tan.|N.Cot.|N.Cos.|   || 

O 

I 

.52992 

.62487 

1.6003 

.84805 

60 

59 

•53017 

527 

•5993 

789 

2 

041 

568 

•5983 

774 

58 

3 

066 

608 

•5972 

759 

57 

4 

091 

•  649 

•5962 

743 

56 

5 

•53115 

.62689 

1.5952 

.84728 

55 

6 

140 

730 

•594' 

712 

54 

7 

164 

770 

.5931 

697 

53 

8 

189 

811 

.5921 

681 

52 

9 
10 

II 

214 

852 

•59" 

666 

51 
50 

49 

.53238 

.62892 

1.5900 

.84650 

263 

933 

.5890 

635 

12 

28S 

.62973 

.5880 

619 

48 

13 

312 

.63014 

.5869 

604 

47 

14 

337 

055 

•  5859 

S88 

46 

15 

•533^1 

.63095 

1.5849 

•84573 

45 

16 

386 

13b 

•5839 

557 

44 

17 

411 

177 

■5829 

542 

43 

18 

_,AJ6- 

217 

.5818 

526 

42 

19 

20 

21 

460 

258 

.5808 

511 

41 
40 

39 

.53484 

.63299 

1.5798 

•84495 

509 

340 

•5788 

480 

22 

534 

380 

•5778 

464 

38 

23 

558 

421 

.5768 

448 

Zl 

24 

583 

462 

.5757 

433 

36 

25 

•53607 

•63503 

1.5747 

•84417 

35 

26 

632 

544 

•5737 

402 

34 

27 

656 

584 

.5727 

386 

ZZ 

28 

681 

625 

•5717 

370 

32 

29 
30 

31 

705 

666 

•5707 

355 

31 
30 

29 

•53730 

•63707 

1.5697 

•84339 

754 

748 

•S687 

324 

32 

779 

789 

•5677 

308 

28 

ZZ 

804 

830 

•5667 

292 

27 

34 

828 

871 

•5657 

277 

26 

35 

•53853 

.63912 

'•5647 

.84261 

25 

3t> 

877 

953 

•5637 

245 

24 

^Z 

902 

•63994 

.5627 

230 

23 

3^ 

926 

.64035 

•5617 

214 

22 

39 
40 

41 

951 

07b 

.5607 

198 

21 
20 

19 

•53975 

.64117 

1-5597 

.84182 

.54000 

158 

.5587 

167 

42 

024 

199 

.5577 

151 

18 

43 

049 

240 

•5567 

135 

17 

44 

073 

281 

•5557 

120 

16 

45 

•54097 

.64322 

1-5547 

.84104 

15 

4b 

122 

363 

.5537 

088 

14 

47 

146 

404 

.5527 

072 

13 

48 

171 

446 

•5517 

057 

12 

49 
50 

51 

195 

487 

•5507 

041 

II 
10 

9 

.54220 

.64528 

1-5497 

.84025 

244 

569 

.5487 

.84009 

52 

269 

610 

•5477 

.83994 

8 

53 

293 

652 

•5468 

978 

7 

S4 

317 

693 

•5458 

962 

6 

55 

•54342 

•64734 

1.5448 

.83946 

5 

5^ 

36b 

775 

■5438 

930 

4 

57 

391 

817 

.5428 

915 

3 

S8 

415 

858 

.5418 

899 

2 

59 
60 

440 

899 

.5408 

883 

I 
0 

.54464 

.64941 

1-5399 

.83867 

N.Cos. 

N.Cot.|N.Tan.| 

N.Sin.|  '  || 

33' 


t 

|N.Sin.|N.Tan.;N.Cot.|N.Cos.|   1 

0 

I 

•54464 
488 

.64941  1.5399 

.83867 

60 

59 

.64982 

.5389 

851 

2 

513 

.65024 

.5379 

835 

58 

3 

537 

065 

.5369 

819 

57 

4 

561 

106 

.5359 

804 

S6 

5 

.5458b 

.65148 

1-5350 

.83788 

55 

6 

610 

189 

•5340 

772 

54 

7 

635 

231 

.5330 

7S6 

53 

8 

659 

272 

•5320 

740 

52 

9 
10 

II 

683 

314 

•53" 
1.5301 

724 

51 
50 

49 

.54708 

•65355 

.83708 

732 

397 

.5291 

692 

12 

756 

438 

•5282 

676 

48 

13 

781 

480 

.5272 

660 

47 

H 

805 

521 

.5262 

64s 

46 

15 

.54829 

•65563 

1-5253 

.83629 

45 

16 

854 

604 

.5243 

613 

44 

7 

878 

646 

•5233 

597 

43 

18 

902 

688 

•5224 

581 

42 

19 
20 

21 

927 

729 

.5214 

565 

41 
40 

39 

.54951 

.65771 

1.5204 

.83549 

975 

813 

•5195 

533 

22 

.54999 

854 

.5185 

517 

38 

23 

•55024 

896 

•5175 

501 

37 

24 

048 

938 

.5166 

48s 

S6 

25 

•55072 

.65980 

1.5156 

.83469 

35 

2b 

097 

.66021 

•5147 

453 

34 

27 

121 

063 

.5137 

437 

33 

28 

145 

105 

•5127 

421 

32 

29 

30 

31 

ib9 

H7 

.5118 

405 

31 
30 

29 

•55194 

.66189 

1.5108 

.83389 

218 

230 

•5099 

373 

32 

242 

272 

.5089 

356 

28 

ZZ 

266 

314 

.508a 

340 

27 

34 

291    356 

.5070 

324 

26 

35 

-55315 

.66398 

I.qo6l 

.83308 

25 

36 

339 

440 

.5051 

292 

24 

37 

363 

482 

•5042 

276 

23 

38 

388 

524 

•5032 

260 

22 

39 
40 

41 

412 

566 

•5023 

244 

21 
20 

19 

•55436 

.66608 

1^5013 

.83228 

460 

650 

.5004, 

212 

42 

484 

692 

•4994 

195 

18 

43 

509 

734 

.4985 

179 

17 

44 

533 

776 

•4975 

163 

16 

45 

•55557 

.66818 

1.4966 

.83147 

15 

46 

581 

860 

•4957 

131 

14 

47 

605 

902 

•4947 

115 

13 

48 

630 

944 

.4938 

098 

12 

49 
50 

51 

654 

.66986 

.4928 

082 

II 
(0 

9 

.55678 

.67028 

1.4919 

.83066 

702 

071 

.4910 

050 

52 

726 

"3 

.4900 

034 

8 

53 

750 

155 

.4891 

017 

7 

54 

775 

197 

.4882 

.83001 

6 

55 

.55799 

.67239 

1.4872 

.82985 

5 

56 

823 

282 

.4863 

969 

4 

57 

847 

324 

•4854 

953 

3 

58 

871 

366 

.4844 

936 

2 

59 
60 

895 

409 

•4835 

920 

I 
0 

.55919 

.67451  1 

1 .4826  .82904 1 

|n.Cos.| 

N.Cot.  N.Tan.  N.  Sin.| 

t 

^•7^ 


KOP 


34' 


t 

NTsin. 

N.Tan.  N.Cot.|N.Cos. 

0 

.55919 
943 

^6745 1 
493 

1.4826 

.82904 

60 

59 

.4816 

887 

2 

968 

536 

.4807 

871 

58 

3 

.55992 

57« 

.4798 

855 

57 

4 

.560161   620 

.4788 

839 

56 

.S 

.56040 

.67663 

1.4779 

.82822 

55 

6 

064 

705 

.4770 

806 

54 

7 

088 

748 

.4761 

790 

53 

8 

112 

790 

.4751 

773 

52 

9 
10 

II 

^Z^ 

832 

.4742 

757 

51 
50 

49 

.56160 

.67875 
917 

1.4733 

.82741 
•  724 

184 

.4724 

12 

208 

.67960 

.4715 

708 

48 

U 

232 

.68002 

.4705 

692 

47 

H 

256 

045 

.4696 

675 

46 

i.S 

.56280 

.68088 

1.4687 

•82659 

45 

lb 

305 

130 

.4678 

643 

44 

17 

329 

173 

.4669 

626 

43 

i8 

353 

215 

.4659 

610 

42 

19 
20 

21 

377 

258 

.4650 

593 

41 
40 

39 

.56401 

.68301 

1. 464 1 

•82577 

425 

343 

.4632 

561 

22 

449 

3«6 

.4623 

544 

38 

23 

473 

429 

.4614 

528 

37 

24 

497 

471 

.4605 

5" 

36 

25 

.56521 

.68514 

1.459b 

.82495 

35 

26 

545 

557 

.4586 

478 

34 

27 

569 

600 

.4577 

462 

33 

28 

593 

642 

.4568 

446 

32 

29 

30 

31 

617 

685 

•4559 

429 

31 
30 

29 

.56641 

.68728 

1.4550 

.82413 

665 

771 

•4541 

396 

32 

689 

814 

•4532 

380 

28 

33 

713 

857 

•4523 

363 

27 

34 

736 

900 

.4514 

347 

26 

35 

.56760 

.68942 

1.4505 

.82330 

25 

3^ 

784 

.68985, 

.4496 

314 

24 

^Z 

808 

.69028 

.4487 

297 

23 

3« 

«32 

071 

.4478 

281 

22 

39 
40 

41 

856 

114 

.4469 

264 

21 
20 

^9 

.56880 

.69157 

1.4460 

.82248 

904 

200 

.4451 

231 

42 

928 

243 

.4442 

214 

18 

43 

952 

286 

•4433 

198 

17 

44 

.56976 

329 

.4424 

181 

16 

45 

.57000 

.69372 

1.4415 

.82165 

15 

4b 

024 

416 

.440b 

148 

14 

47 

047 

459 

.4397 

132 

13 

48 

.071 

502 

.4388 

"5 

12 

49 
50 

51 

095 

545 
.69588 

•4379 
1-4370 

098 
.82082 

11 
10 

9 

•57^19 
143 

631 

.4361 

065 

52 

167 

675 

.4352 

048 

8 

53 

191 

718 

.4344 

032 

7 

54 

215 

761 

•4335 

.82015 

6 

55 

•5723^ 

.69804 

1.4326 

.81999 

5 

5^) 

262 

847 

.4317 

982 

4 

57 

286 

891 

.4308 

965 

3 

5^ 

310 

934 

.4299 

949 

2 

59 
60 

334 

.69977 

.4290 

932 
.81915 

I 
0 

.57358 

.70021 

1.4281- 

N.Cos. 

N.  Cot. 

N.Tan.|N.Sin. 

/ 

36° 

89 

/ 

N.Sin.  N.Tan.  N.Cot.|N. Cos. 

~ 

0 

.57358 

.70021  :  1.4281 

.81915 
899 

60 

59 

38i 

064 

•4273 

2 

405 

107 

.4264 

882 

58 

3 

429 

/151 

•4255 

865 

57 

4 

453 

1^4 

.4246 

848 

56 

5 

.57477 

.70238 

1-4237 

.81832 

55 

6 

501 

281 

.4229 

8iS 

54 

7 

524 

325 

.4220 

798 

S3 

8 

548 

368 

.4211 

782 

52 

9 
10 

572 

412 

.4202 

765 

51 
50 

49 

•57596 

.70455 

i.4»93 

.81748 

619 

499 

.4185 

73» 

12 

643 

542 

.4176 

7H 

48 

13 

667 

586 

.4167 

698 

47 

14 

691 

629 

.4158 

681 

46 

15 

.57715 

.70673 

1.4150 

.81664 

45 

16 

738 

717 

.4141 

647 

44 

17 

762 

760 

.4132 

631 

43 

18 

786 

804 

.4124 

614 

42 

19 
20 

21 

810 

848 

.4115 

597 
.81580 

41 
40 

39 

•57833- 
857 

.70891 

1.4106 

935 

.4097 

563 

22 

881 

.70979 

.4089 

546 

38 

23 

904 

.71023 

.4080 

530 

37 

24 

928 

066 

.4071 

5^3 

36 

25 

•57952 

.71110 

1.4063 

.81496 

35 

26 

976 

154 

•4054 

479 

34 

27 

.57999 

198 

.4045 

462 

33 

28 

.58023 

242 

.4037 

445 

32 

29 
30 

31 

047 

285 

.4028 

428 

31 
30 

29 

.58070 

•71329 

1.4019 

.81412 

094 

373 

.4011 

395 

32 

n8 

417 

.4002 

378 

28 

33 

141 

461 

•3994 

361 

27 

34 

165 

505 

.3985 

344 

26 

35 

.58189 

.71549 

1.3976 

.81327 

25 

36 

212 

593 

.3968 

310 

24 

37 

236 

637 

•3959 

293 

23 

38 

260 

681 

.  .3951 

276 

22 

39 
40 

41 

283 

725 

.3942 

259 

21 
20 

19 

.58307 

.71769 

1.3934 

.81242 

330 

813 

•3925 

225 

42 

354 

857 

.3916 

2C8 

18 

43 

378 

901 

.3908 

191 

17 

44 

401 

946 

•3899 

174 

16 

45 

.58425 

.71990 

1.3891 

.81157 

15 

46 

449 

.72034 

.3882 

140 

H 

47 

472 

078 

-3874 

123 

13 

48 

496 

122 

.386S 

106 

12 

49 
50 

51 

519 

167 

.3857 

089 

10 

9 

•58543 

.72211 

1.3848 

.81072 

567 

255 

.3840, 

055 

52 

590 

299 

•383I' 

038 

8 

53 

614 

344 

.3823 

021 

7 

54 

637 

388 

.3814 

.81004 

6 

55 

.58661 

.72432 

1.3806 

.80987 

5 

56 

684 

477 

•  -3798 

970 

4 

57 

708 

521 

•3789 

953 

3 

58 

731 

565 

•3781 

936 

2 

59 
60 

.58779 

610 

.3772 

919 

0 

•72654 

1.3764 

.80902 

1 

N.Cos. 

N.  Cot.  N.Tan.  N.Sin. 

/ 

KK° 


.^A° 


90 

36° 

/ 

N.Sin 

N.Tan.|N.Cot.|N.Cos.|   || 

0 

I 

.58779 

.72654 

1-3764 

.80902 

60 

S9 

802 

699 

•3755 

885 

2 

826 

743 

•3747 

867 

S8 

3 

849 

788 

.3739 

850 

57 

4 

873 

832 

•3730 

833 

S6 

5 

.58896 

.72877 

1.3722 

.80816 

55 

5 

920 

921 

•3713 

799 

54 

7 

943 

.72966 

•3705 

782 

S3 

8 

967 

.73010 

•3697 

765 

S2 

9 
lO 

II 

.58990 

055 

.3688 

748 

51 
50 

49 

.59014  1  .73100 

1.3680 

•80730 

037 

144 

.3672 

713 

12 

061 

189 

•.3663 

696 

48 

13 

084 

234 

.3655 

679 

47 

14 

108 

278 

•3647 

662 

46 

IS 

•59131 

•73323 

1^3638 

.80644 

4S 

16 

154 

368 

•3630 

627 

44 

17 

178 

413 

.3622 

610 

43 

18 

201 

457 

.3613 

593 

42 

19 
20 

21 

225 

502 

.3605 

576 

41 
40 

39 

.59248 

•73547 

1.3597 

:8o558 

272 

592 

.3588 

541 

22 

295 

637 

.3580 

524 

3ii 

23 

318 

681 

•3572 

507 

37 

24 

342 

726 

•3564 

489 

36 

2S 

■59365 

•73771 

1.3555 

.80472 

3S 

26 

389 

816 

•3547 

455 

34 

27 

412 

861 

•3539 

438 

33 

28 

436 

906 

.3531 

420 

32 

29 

30 

SI 

459 

951 

.3522 

403 

31 

30 

1 
29 

.59482 

•73996 

1.3514 

.80386 

S06 

.74041 

.3506 

36S 

32 

529 

086 

.3498 

351 

28 

33 

552 

131 

.3490 

334 

27 

34 

576 

176 

.3481 

0  ^'^ 

26 

3S 

•59599 

.74221 

1.3473 

.80299 

25 

36 

622 

267 

•3465 

282 

24 

37 

646 

312 

.3457 

264 

23 

3« 

669 

357 

.3449 

247 

22 

39 
40 

41 

693 

402 

.3440 

230 

21 
20 

19 

.59716 

•74447 

1.3432 

.80212 

739    492 

•3424 

195 

42 

763 

S38 

.3416 

178 

18 

43 

786 

583 

.3408 

160 

17 

44 

809 

628 

.3400 

143 

16 

4S 

.59832 

.74674 

1.3392 

.80125 

15 

46 

856 

719 

.3384 

108 

14 

47 

879 

764 

•3375 

091 

13 

48 

902 

810 

•3367 

073 

12 

49 
50 

SI 

926 

855 

•3359 

056 

II 
10 

9 

•59949 

.74900 

1-3351 

.80038 

972 

946 

•3343 

021 

S2 

.59995 

.74991 

•3335 

.Soqps 

8 

53 

.60019 

•75037 

-3327 

.79986 

7 

S4 

042 

082 

•3319 

968 

6 

55 

.60065 

.75128 

i^33ii 

•79951 

5 

5^ 

089 

173 

•3303 

934 

4 

S7 

112 

219 

.3295 

916 

3 

S8 

135 

264 

•3287 

899 

2 

59 
60 

158 

310 

•3278 

881 

0 

.60182 

•75355 

1.3270 

.79864 

N.Cos.  N.Cot.N.Tan.j  N.  Sin.| 

/ 

37' 


'  |N.Sin.  N.Tan. 

N.  Cot! N.  Cos 



0 

I 

.60182 

.75355 

1.3270 

.79864 

60 

S9 

205 

401 

.3262 

846 

2 

228 

447 

•3254 

829 

S8 

J 

251 

492 

.3246 

^  811 

57 

4 

274 

538 

.3238 

793 

S6 

5 

.60298 

•75584 

1.3230 

.79776 

ss 

6 

321 

b29 

.3222 

758 

54 

7 

344 

675 

.3214 

741 

S3 

8 

367 

721 

.3206 

723 

S2 

9 
10 

II 

390 

767 

.3198 

706 

51 
50 

49 

.60414 

.75812 

1.3190 

.79688 

437 

858 

.3182 

671 

12 

460 

904 

•3175 

6S3 

48 

13 

483 

950 

.3167 

635 

47 

14 

S06 

•75996 

.3159 

618 

46 

15 

.60529 

.76042 

1.3151 

.79600 

4S 

16 

553 

088 

.3143 

583 

44 

17 

576 

'§^ 

•3135 

565 

43 

18 

599 

180 

.3127 

547 

42 

19 
20 

21 

622 

226 

.3119 

530 

41 
40 

39 

.60645 

.76272 

1.3111 

.79512 

668 

318 

.3103 

494 

22 

691 

364 

•3095 

477 

38 

23 

714 

410 

•3087 

459 

37 

24 

738 

456 

•3079 

441 

36 

25 

.60761 

.76502 

1.3072 

.79424 

3S 

26 

784 

548 

.3064 

406 

34 

27 

807 

594 

•3056 

388 

33  ■ 

28 

830 

640 

•3048 

371 

32 

29 
30 

31 

853 

686 

•3040 

353 

31 
30 

29  1 

.60876 

•76733 

1.3032 

•79335 

899 

779 

.3024 

318 

32 

922 

825 

•3017 

300 

28  1 

33 

945 

871 

•  .3009 

282 

27 

34 

968 

918 

.3001 

264 

26 

35 

.60991 

•76964 

1.2993 

.79247 

25 

36 

.61015 

.77010 

•2985 

229 

24 

^l 

038 

057 

•2977 

211 

23 

38 

061 

103 

.2970 

193 

22 

39 
40 

41 

084 

149 

.2962 

176 

21 
20 

19 

.61107 

.77196 

1.2954 

•79158 

130 

242 

.2946 

140 

42 

153 

289 

.2938 

122 

18 

43 

176 

335 

.2931 

105 

17 

44 

199 

382 

•2923 

0S7 

16 

45 

.61222 

•77428 

1-2915 

.79069 

15 

46 

245 

475 

.2907 

051 

14 

47 

268 

521 

.2900 

033 

13 

48 

291 

568 

.2892 

.79016 

12 

49 
50 

51 

314 

615  1 

.2884 

.78998 

II 
10 

9 

•61337 

.77661  1 

1.2876 

.78980 

360 

708  1 

•  .2869 

962 

S2 

383 

754! 

.2861 

944 

8 

53 

406 

80.1 

-2853 

926 

7 

S4 

429 

848  1 

.2846 

908 

6 

55 

.61451 

•77895^ 

1.2838 

.78891 

5 

56 

474 

941  ■■ 

.2830 

873 

4 

S7 

497 

.77988  ! 

.2822 

855 

3 

58 

520 

•78035 

.2815 

837 

2 

59 
60 

543 

082  1 

.2807 

819 

0 

.61566 

•78129  1 

1.2799 

.78801 

N.Cos.N.Cot.l 

N.Tan.|  N.  Sin.| 

/ 

K.q^ 


.^9/ 


3 

8° 

39° 

91 

/ 

N.Sin. 

N.Tan. 

N.Cot.|N.Cos. 

' 

N.  Sin.  N.Tan.  N.  Cot.  N.  Cos. 

0 

I 

2 

3 

.61566 

.78129 

1.2799 

.78801 

60 

59 
58 
57 

0 

2 

3 

.62932 

.80978 

1.2349 

•77715 
696 
678 
660 

60 

59 
58 
57 

589 
612 

635 

175 
222 

269 

.2792 
.2784 
.2776 

783 
765 
747 

955 
.62977 
.63000 

.81027 
075 
123 

•2342 
•2334 
•2327 

4 

I 

658 

.61681 

704 

316 

.78363 

410 

.2769 

1. 2761 

.2753 

729 

.78711 

694 

56 

55 
54 

4 
5 
6 

022 

.63045 

068 

171 

.81220 

268 

.2320 

1.2312 

.2305 

641 

•77623 

605 

56 
55 
54 

7 
8 

9 

iO 

II 

12 
13 

749 

772 

--  457 
504 

551 

.2746 
.2738 
•2731 

676 
658 
640 

53 
52 
51 
50 

49 
48 
47 

I 

9 
10 

II 
12 
13 

090 
113 
135 

316 
364 
413 

.2298 
.2290 
.2283 

586 
568 
550 

53 
52 
51 
50 

48 

47 

.61795 

.78598 

1.2723 

.78622 

.63158 

.81461 

1.2276 

.77531 

818 
841 
864 

645 
692 

739 

.2715 
.2708 
.2700 

604 
586 
568 

180 
203 

225 

5'° 

558 
606 

.2268 
.2261 
.2254 

513 
494 
476 

14 

16 

887 

.61909 

932 

786 

•78834 
881 

.2693 

1.2685 

.2677 

550 

•78532 

514 

46 
45 
44 

14 

15 
16 

248 

.63271 

293 

655 
.81703 

752 

.2247 

1.2239 

.2232 

458 

•77439 

421 

46 
45 
44 

17 
18 

J9 
20 

21 
22 
23 

955 
.61978 
.62001 

928 

.78975 
.79022 

.2670 
.2662 
.2655 

496 
478 
460 

43 
42 

41 

40 

39 

2,1 

:^ 

19 
20 

21 
22 
23 

361 

800 

849 
898 

.2225 
.2218 
.2210 

402 
384 
366 

43 
42 
41 
40 

39 
38 
37 

.62024 

.79070 

1.2647 

.78442 

•63383 

.81946 

1.2203 

.77347 

046 
069 
092 

117 
164 
212 

.2640 
.2632 
.2624 

424 
405 

387 

406 
428 
451 

.81995 

.82044 

092 

.2196 
.2189 
.2181 

329 
310 

292 

24 

.62138 
160 

259 

.79306 

354 

.2617 

1.2609 

.2602 

369 
•78351 

36 
35 
34 

24 
25 
26 

473 

.63496 

518 

141 

.82190 
238 

.2174 

1. 2167 

.2160 

273 

.77255 

236 

36 
35 
34 

^1 
28 
29 

30 

31 
32 
33 

183 
206 
229 

401 

449 
496 

.2594 
.2587 
•2579 

315 

297 

279 

3Z 
32 
31 
30 

29 
28 
27 

27 
28 

29 

30 

31 

32 

540 
563 

585 

287 
336 
38? 

•2153 
.2145 
.2138 

218 

199 
181 

33 
32 
31 
30 

29 

28 
27 

.62251 

.79544 

1.2572 

.78261 

.63608 

.82434 

1.2131 

.77162 

274 
297 
320 

639 
686 

.2564 
•2557 
.2549 

243 
225 

630 
675 

483 

.2124 
.2117 
.2109 

144 
125 
107 

34 

342 

734 

.79781 

829 

.2542 

1.2534 

.2527 

188 

.78170 

152 

26 
25 
24 

34 
35 
36 

698 

.63720 

742 

629 
.82678 

727 

.2102 

1.2095 

.2088 

088 

.77070 

051 

26 

25 
24 

37 
38 
39 
40 

41 

42 

43 

411 
433 

456 

877 

924 

.79972 

.2519 
.2512 
.2504 

134 
116 

098 

23 
22 
21 

20 

19 
18 

17 

37 
38 
39 
40 

41 

42 
43 

765 
787 
810 

776 
825 
874 

.2081 
.2074 
.2066 

033 
.77014 
.76996 

23  ' 

22 

21 

20 

19 
18 

17 

.62479 

.80020 

1.2497 

.78079 

•63832 

.82923 

1.2059 

.76977 

502 
524 
547 

067 

^\^ 
163 

.2489 
.2482 
.2475 

061 

043 
025 

854 
899 

.82972 

.83022 

071 

.2052 
.2045 
.2038 

959 
940 
921 

44 
45 
46 

.62592 
615 

211 

.80258 
306 

.2467 

1.2460 

.2452 

.78007 

•77988 

970 

16 

15 
14 

44 
45 
46 

922 

•63944 
966 

120 

.83169 

218 

.2031 

1.2024 

.2017 

903 

.76884 

866 

16 
15 
14 

% 

49 
50 

51 

52 
53 

638 
660 
683 

354 
402 
450 

•2445 
•2437 
.2430 

952 
934 
916 

13 
12 
II 
10 

9 
8 

7 

47 
48 
49 
50 

51 

52 
53 

.63989 
.64011 

033 

268 

317 

366 

.2009 
.2002 
.1995 

847 
828 
810 

13 
12 

10 

9 
8 

7 

.62706 

.80498 

1.2423 

.77897 

.64056 

.83415 

1. 1988 

.76791 

728 
751 
774 

546 
594 
642 

•2415 
.2408 
.2401 

879 
861 

843 

078 
100 
123 

465 
564 

.1981 

.1974 
.1967 

772 
754 
735 

54 

796 

.62819 

842 

690 

.80758 

786 

•2393 
1.2386 

.2378 

824 

.77806 

788 

6 

5 
4 

54 

145 

.64167 

190 

613 

.83662 

712 

.i960 

1. 1953 
.1946 

717 
.76698 

679 

6 

5 
4 

59 
60 

864 
887 
909 

834 
882 
930 

.2371 
.2364 

.2356 

769 
751 
733 

3 
2 
I 

0 

57 
58 
59 
60 

212 

234 
256 

860 

.1939 
.1932 

.1925 

661 
642 
623 

3 

2 

0 

.62932 

.80978 

1.2349 

•77715 

.64279 

.83910 

1.1918 

.76604 

N.Cos.|N.Cot. 

;N.Tan.|  N.Sin. 

/ 

N.Cos.  N.  Cot.lN.Tan.i  N.  Sin. 

r 

Kt 


^(\' 


92 

40° 

1 

N.  Sin.|N.Tan 

iN.Cot.|N.Cos. 

0 

.64279 

.83910 

1  1.1918  .76604 

60 

59 

301 

.83900 

.1910 

586 

2 

323 

.84009 

.1903 

S67 

S8 

3 

346 

059 

.1896 

548 

57 

4 

368 

108 

.1889 

530 

56 

S 

.64390 

.84158 

1. 1882 

.76511 

55 

6 

412 

208 

.1875 

492 

54 

7 

435 

258 

..1868 

473 

53 

8 

457 

307 

.1861 

455 

52 

9 
10 

II 

479 

357 

.1854 

436 

51 
50 

49 

.64501 

.84407 

1.1847 

.76417 

524 

457 

.1840 

398 

12 

546 

507 

.1833 

380 

48 

13 

568 

556 

.1826 

361 

47 

14 

590 

606 

.1819 

342 

46 

15 

.64612 

.84656 

1.1812 

•76323 

45 

lb 

^35 

706 

.1806 

304 

44 

17 

657 

7S6 

.1799 

286 

43 

18 

679 

806 

.1792 

267 

42 

19 
20 

21 

701 

856 

.1785 

248 

41 
40 

39 

.64723 

.84906 

1.1778 

.76229 

746 

.84956 

.1771 

210 

22 

768 

.85006 

.1764 

192 

38 

23 

790 

057 

•1757 

173 

37 

24 

812 

107 

.i7?o 

154 

36 

25 

.64834 

•85157 

1-1743 

•76135 

35 

2b 

856 

207 

•1736 

lib 

34 

^Z 

878 

257 

.1729 

097 

33 

28 

901 

308 

.1722 

078 

32 

29 

30 

3^ 

923 

358 

.1715 

059 

31 
30 

29 

.64945 

.85408 

1.1708 

.76041 

967 

458 

.1702 

022 

32 

.64989 

509 

.1695 

.76003 

28 

33 

.65011 

559 

.1688 

.75984 

27 

34 

033 

609 

.1681 

965 

26 

35 

•65055 

.S5660 

1. 1674 

.75946 

25 

3^ 

077 

710 

.1667 

927 

24 

37 

100 

761 

.1660 

908 

23 

3« 

122 

811 

•1653 

889 

22 

39 
40 

41 

144 

862 

.1647 

870 

21 
20 

19 

.65166 

.85912 

1. 1640 

.75S51 

188 

.85963 

.1633 

832 

42 

210 

.86014 

.1626 

813 

18 

43 

232 

064 

.1619 

794 

»7 

44 

254 

"^ 

.1612 

775 

16 

45 

•65276 

.86166 

1. 1606 

•W756 

15 

4b 

298 

216 

.1599 

738 

14 

47 

320 

267 

.1592 

719 

13 

48 

342 

318 

.1585 

700 

12 

49 
50 

51 

3<34 

368 

.1578 

680 

II 
10 

9 

.65386 

.86419 

^•i57i 

.75661 

408 

470 

.1565 

642 

52 

430 

521 

.15.58 

623 

8 

53 

452 

572 

•1551 

604 

7 

54 

474 

623 

•1544 

58s 

6 

55 

.65496 

.86674 

1-1538 

.75566 

5 

5^ 

518 

725 

.1531 

547 

4 

57 

540 

776 

.1524 

528 

3 

5^ 

562 

827 

•1517 

509 

2 

59 
60 

584 

878 

.1510 

490 

I 
0 

.65606 

.86929 

1. 1504 

•75471 

N.Cos. 

N.  Cot. 

N.Tan.j  N.  Sin. 

' 

41° 

'  iN.Sin.jN.Tan. 

N.  Cot.] N.Cos. 

1 

0 

.65606 

.86929 

1. 1504 

.75471 

60 

S9 

628 

.86980 

.1497 

452 

2 

650 

.87031 

.1490 

433 

58 

3 

672 

082 

.1483 

414 

57 

4 

694 

0  '^3 

.1477 

395 

S6 

.  5 

.65716 

.87184 

1. 1470 

•75375 

55 

b 

738 

236 

>  .1463 

356 

54 

7 

759 

287 

.1456 

337 

53 

8 

781 

338 

.1450 

318 

52 

9 
10 

II 

803 

389 

•H43 

299 

51 
50 

49 

.65825 

.87441 

1.1436 

.75280 

847 

492 

.1430 

261 

12 

869 

543 

.1423 

241 

48 

13 

891 

595 

.1416 

222 

47 

14 

913 

646 

.1410 

203 

46 

15 

•65935 

.87698 

1. 1403 

.75184 

45 

lb 

956 

749 

•1396 

165 

44 

17 

.65978 

801 

.1389 

146 

43 

18 

.66000 

852 

.1383 

126 

42 

19 
20 

022 

904 

.1376 

107 

41 
40 

.66044 

•87955 

1. 1369 

.75088 

21 

066 

.88007 

.1.363 

069 

39 

22 

088 

059 

.1356 

050 

38 

23 

109 

no 

.1349 

030 

37 

24 

131 

162 

.1343 

•75011 

36 

25 

.66153 

.88214 

1.1336 

.74992 

35 

2b 

175 

265 

.1329 

973 

34 

27 

197 

317 

.1323 

953 

33 

28 

218 

369 

.1316 

934 

32 

29 
30 

31 

240 

421 

.1310 

915 

31 
30 

29 

.66262 

.88473 

1.1303 

.74896 

284 

524 

.1296 

876 

32 

30b 

576 

.1290 

8S7 

28 

33 

327 

628 

.1283 

838 

27 

34 

349 

680 

.1276 

818 

26 

35 

•66371 

.88732 

1. 1 270 

•74799 

25 

36 

393 

784 

.1263 

780 

24 

37 

414 

836 

.1257 

760 

23 

38 

436 

888 

.1250 

741 

22 

39 
40 

41 

458 

940 

.1243 

722 

21 
20 

19 

.66480 

.88992 

1. 1 237  1.74703 

501 

.89045 

.1230 

683 

42 

523 

097 

.1224 

664 

18 

43 

545 

149 

.1217 

644 

17 

44 

566 

201 

.1211 

625 

16 

45 

.66588 

•89253 

1.1204 

.74606 

15 

46 

610 

306 

.1197 

586 

14 

47 

632 

358 

.1191 

567 

13 

48 

653 

410 

.1184 

548 

12 

49 
50 

SI 

675 

463 

.1178 

528 

II 
10 

9 

.66697 

718 

.89515 
567 

1.1171 

.74509 

.1165 

489 

52 

740 

620 

.1158 

470 

8 

53 

762 

672 

.1152 

451 

7 

54 

783 

725 

.1145 

431 

6 

55 

.66805 

•89777 

I. "39 

•74412 

5 

56 

827 

830 

.1132 

392 

4 

57 

848 

883 

.1126 

373 

3 

58 

870 

935 

.1119 

353 

2 

59 
60 

891 

.89988 

.1113 

334 

0 

.66913 

.90040 

I.I  106 

.74314 

N.Cos. 

N.Cot. 

N.Tan.j  N.  Sin. 

^ 

AQ' 


A»° 


42' 


'  iN.Sin. 

N.Tan^ 

NTcot. 

N^  Cos 

1 

0 

I 

.66913 

.90040 

I.I  106 

.743 « 4 

60 

59 

935 

093 

.1100 

295 

2 

95^ 

146 

.1093 

276 

S8 

3 

978 

199 

.1087 

256 

57 

i  -* 

.66999 

251 

.1080 

237 

56 

1  S 

.67021 

.90304 

1.1074 

.74217 

55 

0 

043 

357 

.1067 

198 

54 

7 

064 

410 

.1061 

178 

53 

8 

086 

463 

.1054 

159 

52 

9 
10 

IL 

107 

5.0 

.1048 

139 

51 
50 

49 

.67129 

.90569 

1.104 1 

.7412a 

151 

621 

•1035 

100 

12 

172 

674 

.1028 

080 

48 

13 

194 

727 

.1022 

061 

47 

H 

215 

781 

.1016 

041 

46 

IS 

.67237 

.90834 

1. 1009 

.74022 

45 

i6 

258 

887 

.1003 

.74002 

44 

17 

280 

940 

.0996 

•73983 

43 

18 

301 

.90993 

.0990 

963 

42 

19 
20 

21 

323 

.91046 

.0983 

944 

41 
40 

39 

.67344 

.91099 

1.0977 
.0971 

•73924 

300 

153 

904 

22 

3«7 

206 

.0964 

885 

3>i 

23 

409 

259 

.0958 

865 

37 

24 

430 

3^3 

.0951 

846 

36 

2S 

.67452 

.91366 

1.0945 

.73826 

35 

26 

473 

419 

•0939 

806 

34 

27 

495 

473 

.0932 

787 

33 

28 

516 

526 

.0926 

767 

32 

29 
30 

31 

53« 

580 

.0919 

747 

31 
30 

29 

•67559 

•91633 

1.0913 

•73728 

580 

687 

.0907 

708 

32 

602 

740 

.0900 

688 

28 

33 

623 

794 

.0894 

669 

27 

34 

645 

847 

.0888 

,649 

26 

3S 

.67660 

.91901 

1. 088 1 

.73629 

25 

36 

688 

•91955 

.0875 

610 

24 

37 

709 

.92008 

.0869 

590 

23 

3S 

730 

062 

.0862 

570 

22 

39 
40 

41 

752 

116 

.0856 

551 

21 
20 

19 

•67773 

.92170 

1.0850 

•73531 
511 

795 

224 

.0843 

42 

.  816 

277 

.0837 

491 

18 

43 

^37 

331 

.0831 

472 

17 

44 

859 

385 

.0824 

452 

16 

45 

.67880 

.92439 

1.0818 

•73432 

15 

46 

901 

493 

.0812 

413 

14 

47 

923 

547 

.0805 

393 

13 

48 

944 

601 

.0799 

373 

12 

49 
50 

SI 

965 

655 

.0793 

353 

II 
10 

9 

.67987 

.92709 

1.0786 

■73333 

.68008 

763 

.0780 

314 

S2 

029 

817 

•0774 

294 

8 

53 

051 

872 

.0768 

274 

7 

S4 

072 

926 

.0761 

.  254 

6 

55 

.68093 

.92980 

^•0755 

•73234 

5 

5^ 

"5 

•93034 

•0749 

215 

4 

S7 

1.36 

088 

.0742 

195 

3 

5« 

157 

H3 

.0736 

175 

2 

59 
80 

179 

197 

.0730 

155 

0 

.68200 

•93252 

1.0724 

•73135 

N.Cos.  N.Cot. 

N.Tan. 

N.Sin.l  '  || 

43° 

93 

I'    |N.  Sin.  N.Tan.  N.Cot.  N.Cos.|    l| 

0 

I 

.68200 
221 

.93252  1.0724 

•73'35 

60 

59 

306   .0717 

116 

2 

242 

360   .0711 

096 

S8 

3 

264 

4'5 

•0705 

076 

57 

4 

285 

469 

.0699 

056 

S6 

5 

.68306 

•93524 

1.0692 

•73036 

ss 

6 

327 

578 

.0686 

.73016 

54 

7 

349 

633 

.0680 

.72996 

S3 

8 

370 

688 

.0674 

976 

52 

9 
10 

II 

391 

742 

.0668 

957 

51 
50 

49 

.68412 

•93797 

1. 066 1 

•72937 

434 

852 

•0.655 

917 

12 

455 

906 

.0649 

897 

48 

13 

476 

.93961 

.0643 

877 

47 

H 

497 

.94016 

•0637 

8S7 

46 

15 

.68518 

.94071 

1.0630 

•72837 

4S 

16 

539 

125 

.0624 

817 

44 

17 

^t' 

180 

.0618 

797 

43 

18 

582 

235 

.0612 

777 

42 

19 
20 

21 

603 

290 

.0606 

757 

41 
40 

39 

.68624 

.94345 

1.0599 

•72737 

645 

400 

•0593 

717 

22 

666 

455 

.0587 

697 

38 

23 

688 

510 

•0581 

677 

37 

24 

709 

565 

.0575 

657 

36 

25 

.68730 

.94620 

1.0569 

.72637 

35 

26 

751 

676 

.0562 

617 

34 

27 

772 

731 

.0556 

597 

33 

28 

793 

786 

•0550 

577 

32 

29 
30 

31 

814 

841 

.0544 

557 

31 
30 

29 

•68835 

.94896 

1^0538 

.72537 

857 

•94952 

•0532 

517 

32 

878 

•95007 

.0526 

497 

28 

33 

899 

062 

.0519 

477 

27 

34 

920 

118 

.0513 

457 

26 

35 

.68941 

•95173 

1.0507 

•72437 

25 

36 

962 

229 

•0501 

417 

24 

37 

.68983 

284 

•0495 

397 

23 

3^ 

.69004 

340 

.0489 

377 

22 

39 
40 

41 

025 

395 

•0483 

357 

21 
20 

19 

.69046 

.95451  1.0477 

•72337 

067 

506 

.0470 

317 

42 

088 

562 

.0464 

297 

18 

43 

109 

618 

•0458 

277 

17 

44 

130 

673 

.0452 

257 

16 

45 

.69151 

•95729 

1.0446 

.72236 

15 

46 

172 

785 

.0440 

216 

14 

47 

193 

841 

.0434 

196 

13 

48 

214 

897 

.0428 

176 

12 

49 
50 

51 

235 

.95952 

.0422 

156 

II 
10 

9 

.69256 

.96008 

1. 041 6 

.72136 

277 

064 

.0410 

n6 

52 

298 

120 

.0404 

095 

8 

53 

319 

176 

.0398 

075 

7 

54 

340 

232 

•0392 

055 

6 

55 

.69361 

.96288 

1.0385 

•72035 

5 

56 

382 

344 

.0379 

.72015 

4 

57 

403 

400 

.0373 

•71995 

3 

58 

424 

457 

•0367 

974 

2 

59 
60 

445 

513 

•0361 

954 

I 
0 

.69466 

.96569  1  1.0355 

•71934 

_.„ 

N.Cos.  N.  Cot.  N.Tan.;  N.  Sin.| 

r    1 

AT 


Aa^ 


94 


44" 


/ 

N.Sin.l 

N.Tan.iN.Cot.JN.Cos.l   | 

I 

.69466 
487 

.96569 

1-0355  -719341 

60 

59 

625 

•0349 

914 

2 

508 

681 

•0343 

894 

58 

3 

529 

738 

.0337 

873 

57 

4 

549 

794 

.0331 

853 

56 

5 

.69570 

.96850 

i^o325 

.71833 

55 

6 

591 

907 

.0319 

813 

54 

7 

612 

.96963 

•0313 

792 

53 

8 

633 

.97020 

.  -0307 

772 

52 

' 

9 
lO 

II 

654 

076 

.0301 

752 

51 
50 

49 

.69675 

.97133 

1.0295 

.71732 

696 

189 

.0289 

711 

12 

717 

246 

.0283 

691 

48 

U 

737 

302 

.0277 

671 

47 

14 

758 

359 

.0271 

650 

46 

15 

.69779 

.97416 

1.0265 

.7x630 

45 

i6 

800 

472 

.0259 

610 

44 

17 

821 

529 

•0253 

590 

43 

18 

842 

586 

.0247 

569 

42 

19 
20 

21 

862 

643 

.0241 

549 

41 
40 

39 

.69883 

.97700 

1.0235 

.71529 

904 

756 

.0230 

5°? 

22 

925 

813 

.0224 

488 

38 

23 

946 

870 

.0218 

,468 

37 

24 

966 

927 

.0212 

447 

36 

25 

.69987 

.97984 

1.0206 

.71427 

35 

26 

.70008 

.98041 

.0200 

407 

34 

27 

029 

098 

.0194 

386 

33 

28 

049 

155 

.0188 

366 

32 

29 
30 

31 

070 

213 

.0182 

345 

31 
30 

29 

.70091 

.98270 

1. 01 76 

•71325 

112 

327 

.0170 

305 

32 

132 

384 

.0164 

284 

28 

33 

153 

441 

.0158 

264 

27 

34 

174 

499 

.0152 

243 

26 

^1 

•70195 

.98556 

1.0147 

.71223 

25 

36 

215 

613 

.0141 

203 

24 

37 

236 

671 

•0133 

182 

23 

38 

257 

728 

.0129 

162 

22 

39 
40 

41 

277 

786 

.0123 

141 

21 
20 

19 

.70298 

.98843 

1.0117 

.71121 

319 

901 

.0111 

100 

42 

339 

.98958 

.0105 

080 

18 

43 

360 

.99016 

.0099 

059 

17 

44 

381 

073 

.0094 

039 

16 

45 

.70401 

•991 31 

1.0088 

.71019 

15 

46 

422 

189 

.0082 

.70998 

14 

47 

443 

247 

.0076 

978 

13 

48 

463 

304 

.0070 

957 

12 

49 
50 

51 

484 

362   .0064 

937 

II 
10 

9 

•70505 

.99420 

1.0058 

.70916 

525 

478 

.0052 

896 

52 

546 

536 

.0047 

875 

8 

53 

567 

594 

.0041 

855 

7 

54 

587 

652 

•0035 

83.4 

6 

55 

.70608 

.99710 

1.0029 

.70813 

5 

56 

628 

768 

.0023 

793 

4 

57 

649 

826 

.0017 

772 

3 

58 

670 

884 

.0012 

752 

2 

59 
60 

690 

.99942 

.0006 

731 

I 
0 

.70711 

1. 0000  j  1. 0000 

.70711 

|n  Cos.|N.Cot.|N.Tan.!  N.  Sin.|  ' 

A.*i' 


IV.  CIRCULAR  ARCS 

WITH 

R/ 

\I)IUS 

UNITY.    95 

DEGREES. 

1  MINUTES. 

SECONDS. 

O^ 

0.0000000 

60 

61 

1. 047 1 9  76 

120^ 

121 

2.09439  5 » 
2.1118484 

0' ,  o.ooocx)  00 

O",  0.0000000 

0.0174533 

1 .06465  08 

I 

0.0002909 

I 

0.0000048 

2 

0.03490  66 

62 

1.0821041 

122 

2.12930 17 

2 

0.00058  18 

2 

0.0000097 

3 

0.05235  99 

63 

1-09955  74 

123 

2.1467550 

3 

0.00087  27 

3 

0.0000 1  45 

4 

0.06981  32 

64 

1.11701  07 

124 

2.1642083 

4 

0.001 16  36 

4 

0.00001  94 

5 

0.08726  65 

65 

1. 1 3446  40 

125 

2.18166 16 

5 

0.0014544 

5 

0.00002  42 

6 

0.1047 1  9^ 

66 

1.15191  73 

126 

2.1991149 

b 

0.0017453 

6 

0.00002  91 

7 

0.1 22 1 7  30 

67 

1. 1 6937  06 

127 

2.2165682 

7 

0.00203  62 

7 

0.00003  39 

8 

0.13962  63 

68 

1. 1 8682  39 

128 

2.23402 14 

8 

0.00232  71 

8 

0.00003  88 

9 
10 

II 

0.1570796 

69 
70 

71 

1.20427  72 
1.2217305 

129 
130 

131 

2.2514747 

9 
10 

11 

0.00261  80 

0 

10 

11 

0.00004  36 

0.1745329 

2.26892  80 

0.00290  89 

0.00004  85 
0.00005  33 

0.1919862 

1.23918  38 

2.2863813 

0.0031998 

12 

0.20943  95 

72 

1.25663  71 

132 

2.30383  46 

12 

0.00349  07 

12 

0.00005  82 

13 

0.22689  28 

73 

1.2740904 

^33 

2.3212879 

13 

0.00378  15 

13 

0.00006  30 

H 

0.2443461 

74 

1.2915436 

1.34 

2.3387412 

14 

0.00407  24 

M 

0.00006  79 

15 

0.2617994 

75 

1.3089969 

135 

2.356194? 

15 

0.00436  3i 

15 

0.00007  27 

16 

0.27925  27 

76 

1.3264502 

136 

2.37364  78 

16 

0.00465  42 

16 

0.00007  76 

17 

0.2967060 

77 

1-3439035 

1.37 

2.391 10 11 

17 

0.00494  5 1 

17 

0.00008  24 

18 

0.3141593 

78 

1.36135  68 

138 

2.40855  44 

18 

0.00523  60 

18 

0.00008  73 

19 
20 

21 

0.33161  26 

79 
80 

81 

1.37881  01 

139 
140 

141 

2.42600  77 

19 
20 

21 

0.00552  69 

19 
20 

21 

0,00009  21 

0.34906  59 

1.3962634 

2.44346 10 

0.00581  78 

0.00009  70 

0.36651  91 

1.41371  67 

2.46091 42 

0.00610  87 

o.oooio  18 

22 

0.38397  24 

82 

1.4311700 

142 

2.47836  75 

22  1  0.00639  95 

22 

o.oooio  67 

23 

0.40142  57 

83 

1.4486233 

143 

2.49582  08 

23 

0.00669  04 

23 

0.0001 1  15 

24 

0.4188790 

84 

1.46607  66 

144 

2.5132741 

24 

0.00698  13 

24 

0.000 11  64 

25 

0.43633  23 

85 

1.4835299 

HS 

2.53072  74 

25 

0.00727  22 

25 

0.00012  12 

26 

0.45378  56 

86 

1.50098  32 

146 

2.5481807 

26 

0.0075631 

26 

0.00012  61 

27 

0.4712389 

87 

1. 5 1 843  64 

H7 

2.56563  40 

27 

0.00785  40 

27 

0.0001309 

28 

0.48869  22 

88 

1-5358897 

148 

2-58308  73 

28 

0.0081449 

28 

0.0001357 

29 
30 

31 

0.50614  51 

89 
90 

91 

1.5533430 

149 
150 

151 

2.6005406 

29 
30 

31 

0.00843  58 

29 
30 

31 

0.0001406 

0.52359  88 

1-5707963 

2-6179939 

0.00872  66 

0.00014  54 

0.54105  21 

1.5882496 

2.63544  72 

0.00901  75 

0.00015  03 

32 

0.5585054 

92 

1.60570  29 

152 

2.6529005 

32 

0.00930  84 

32 

0.00015  5^ 

33 

0.57595  87 

93 

1.6231562 

153 

2.67035  38 

33 

0.00959  93 

33 

0.0001600 

34 

0.59341  19 

94 

1.6406095 

154 

2.68780  70 

34 

0.00989  02 

34 

0.0001648 

35 

0.61086  52 

95 

1.6580628 

iSS 

2.7052603 

35 

0.01018  u 

35 

0.0001697 

36 

0.62831  85 

96 

1.67551  61 

156 

2.72271 36 

36 

0.01047  20 

36 

0.00017  45 

37 

0.64577  18 

97 

1.6929694 

157 

2.74016  69 

37 

0.01076  29 

37 

0.00017  94 

3« 

0.6632251 

98 

1.7104227 

158 

2.7576202 

38 

0.01105  38 

38 

0.0001842 

39 
40 

41 

0.68067  84 

99 
100 

lOI 

1.72787  60 

159 
160 

161 

2.7750735 

39 
40 

41 

0.0113446 

39 
40 

0.0001891 

0.69813  17 

i.''4532  93 

2.7925268 

0.0116355 

0.00019  39 

0-7155850 

1.7627825 

2.8099801 

0.01 192  64 

41 

0.00019  88 

42 

0.73303  ^3 

102 

1.78023  58 

162 

2.82743  34 

42 

0.01221  73 

42  1  0.00020  36 

43 

0.75049  16 

103 

1.79768  91 

163 

2.8448867 

43 

0.0125082 

43  0.00020  85 

44 

0.7679449 

104 

1.81514  24 

164 

2.86234  00 

44 

0.0127991 

44  0.00021  33 

45 

0.78539  82 

105 

1-8325957 

i6s 

2.87979  33 

45 

0.0130900 

45  0.00021  82 

45 

0.80285  15 

106 

1.8500490 

166 

2.89724  66 

46 

0.0133809 

46  i  0.00022  30 

47 

0.82030  47 

107 

1.8675023 

167 

2.91469  99 

47 

0.01367  17 

47  1  0.00022  79 

48 

0.83775  80 

108 

1.88495  56 

168 

2.93215  31 

48 

0.0139626 

48  1  0.00023  27 

49 
50 

51 

0.85521  13 

109 
110 
m 

1.9024089 

169 
170 

171 

2.94960  64 

49 
50 

51 

0.01425  35 

49 
50 

0.00023  76 

0.8726646 

1.91986  22 

2.96705  97 

0.0145444 

0.00024  24 

0.89011  79 

I-9373I  55 

2.98451  30 

0.01483  53 

51 

0.00024  73 

52 

0.90757  12 

112 

1.9547688 

172 

3.0019663 

52 

0.01512  62 

52 

0.00025  21 

53 

0.92502  45 

"3 

1.97222  21 

^73 

3.01941  96 

53 

0.01541  71 

53 

0.00025  70 

54 

0.94247  78 

114 

1-9896753 

174 

3.03687  29 

54 

0.0157080 

54 

0.00026  18 

55 

0.95993" 

115 

2.00712  86 

17s 

3.05432  62 

55 

0.0159989 

55  0.00026  66  II 

5^ 

0.9773844 

116 

2.02458  19 

176 

30717795 

56 

0.0162897 

56 

0.00027  15 

57 

0.99483  77 

117 

2.0420352 

177 

3.08923  28 

57 

0.0165806 

57 

0.00027  63 

5« 

1.01229  10 

118 

2.05948  85 

178 

3.10668  61 

58 

0.01687  '5 

58 

0.00028  12 

59 
60 

1.0297443 

119 
120 

2.07694  18 

179 
180 

3-1241394 

59 
60 

0.01716  24 

59 
60 

0.00028  60 

1. 047 1 9  76 

2.0943951 

3.1415927 

0.0174533 

0.00029  09 

DE 

IGREES. 

MINUTES. 

SECONDS.  i| 

96 

V. 

CONVERSION 

OF 

LOGARITHMS. 

Base  of  common  logarithms 

=  10.                 1 

Base  of  Naperian  logarithms  (<?)           =  2.71828  18284  59045  23536 
Com.  Log.  e    =  Jlf  (Modulus  of  Com.  Logs.)  =  0.43429  44819  03251  82765 

Nap.  Log.  10  =  — 

=  2.30258  50929  94045  68402 

Com.  Log.  N  =  jV  X  Nap.  Log.  A'. "" 
Nap.  Log.  JV  =—  X  Com.  Log.  N. 

►  where  iV  denotes  any  number. 

Multiples  of  M. 

Multiples  of  J  . 
M 

0 

2 

3 

0.00000  000 

50 

51 

52 
53 

21.71472  410 
22.14901  858 
22.58331  306 
23.01760754 

0 

I 

2 
3 

0.00000  000 
2.30258  509 
4.60517019 
6.90775J28 

50 

51 

52 
53 

115.12925465 

0.43429  448 
0.86858  896 
1.30288345 

117.43183974 

119-73442484 
122.03700993 

4 

I 

1-73717793 
2.17147241 
2.60576  689 

54 

11 

23.45190202 
23.88619  6^0 
24.32049  099 

4 
5 
6 

9.21034037 
1 1. 5 1 292  546 
13.81551056 

54 
55 
56 

124.33959502 
126.64218011 
128.94476521 

9 
10 

II 

12 

13 

3.04006  137 

3-47435  586 
3.90865  034 

57 
58 
59 
60 

61 
62 
63 

24-75478  547 
25.18907995 
25.62337  443 

I 

9 
10 

II 
12 
13 

1 6. 1 1 809  565 
18.42068074 
20.72326  584 

11 

59 
60 

61 
62 
63 

131.24735030 

133-54993539 
135-85252049 

4.34294  482 

26.05766891 

23.02585  093 

138.15510558 
140.45769067 
142.76027577 
145.06286086 

477723  930 
5-21153378 
5.64582  826 

26.49196340 
26.92625  788 
27.36055  236 

25.32843  602 
27.63102  112 
29.93360621 

6.08012275 
6.51441  723 
6.94871  171 

64 
65 

66 

27.79484  684 
28.22914  132 
28.66343  581 

14 

32.236J9  130 

34-53877  639 
36.84136  149 

64 

147-36544  595 
149.66803  104 
151.97061  614 

19 
20 

21 

22 
23 

7.38300619 
7.81730067 
8.25159  516 

67 
68 

69 

70 

71 

72 

73 

29.09773  029 
29.53202  477 
29.96631  925 

17 
18 

19 

20 

21 
22 
23 

39.14394658 
41.44653  167 
43.74911677 

% 

69 
70 

71 

72 
73 

154.27320123 
15657578632 
158.87837  142 

8.68588  964 

J040o6]tj73_ 
30.83490  822 
31.26920270 
31.70349718 

46.05170  186 

161. 18095  651 

9.12018412 
9.55447  860 
9.98877  308 

48.35428  695 
50.65687  205 
52.95945  714 

163.48354  160 
165.78612670 
168.08871  179 

24 

^1 

10.42306  757 
10.85736  205 
11.29165653 

74 
75 
76 

32.13779  166 
32.57208  614 
33.00638  062 

24 

25 
26 

55.26204  223 
57.56462  732 
59.86721  242 

74 
75 
76 

170.39129688 
172.69388  197 
174.99646707 

27 
28 
29 

30 

3^ 
32 
33 

11.72595  lOI 
12.16024  549 
12.59453998 

77 
78 
79 
80 

81 
82 
83 

33.44067511 
33.87496  959 
34.30926  407 

_3474355  855^ 

35- '7785  303 
35.61214752 
36.04644  200 

27 
28 
29 

1  30 

31 
32 
33 

62.16979751 
64.47238  260 
66.77496  770 

W 

79 
80 

81 
82 
83 

177.29905216 
179.60163725 
181.90422235 

13.02883446 

69.07755  279 

^71.38013788 
73.68272  298 
75.98530  807 

184.20680744 

13.46312894 
13.89742342 
H-33I7I  790 

186.50939253 
188.81197763 
191. 11456  272 

34 
35 
36 

14.76601  238 
15.20030687 
15.63460  135 

84 

85 
86 

36.48073  648 
36.91503096 
3734932  544 

34 
35 
36 

78.28789316 
80.59047821; 
82.89306  ss's 

84 
11 

193.41714781 
i95-7'973  290 
198.02231  800 

37 
3« 
39 
40 

41 
42 

43 

16.06889583 
16.50319  031 
16.93748479 
17.37177928 
17.80607376 
18.24036  824 
18.67466  272 

87 
88 

89 

90 

91 
92 
93 

3778361  993 
38.21791  441 
38.65220  889 

37 
38 
39 
40 

41 
42 
43 

85.19564844 

87.49823  3 S3 
89.800S1  863 

87 
88 
89 
90 

91 
92 
93 

200.32490309 
202.62748818 
204.93007  328 

39.08650  337 

39.52079  785 

39-95509  234 
40.38938  682 

92.10340372 

207.23265  837 

94.40598  881 
9670857  391 
99.01  u  5  900 

209.53524346 
211.83782  8s6 
214.14041  365 

44 
45 
46 

19.10895  720 
19.54325  169 
19-97754617 

94 
95 
96 

40.82368  1 30 
41.25797578 
41.69227026 

44 

IOI.3I374409 
103.61632  918 
I05.9I89I  428 

94 

11 

216.44299874 

218.74558383 
221.04816893 

49 

50 

20.41184065 
20.84613513 
21.28042  961 

97 
98 

99 

100 

42.12656474 
42.56085  923 
42.99515371 

ti 

49 

50 

108.22149937 

110.52408446 
112.82666956 

99 
100 

22335075  402 
225.65333911 
227.95592421 

21.71472410 

43-42944819 

115. 12925  465 

230.25850930 

97 


TRIGONOMETRIC    FORMULAS. 


sm^a  +  cos^a  =  l. 
sec'-^a  =  I  +tan2a. 
cosec^a  =  I  +  cot^a. 

sino  =  db 


QOta 


Vi  +  tan2, 


cos  a 

_  cos^ 

sin  a 

cos  a  —^± 


sec  a 


i_ 

COSO 

I 

sin  a 


sin  {a  ±  (3)  =  sin  a  cos  j3  ±  cos  a  sin  /3. 

cos  ( a  ±  /y)  =  cos  «  cos  /3  =F  sin  a  sin  ^^ 

»      /      ,    «\          f^"  <"■  ±  tan  /3 
,  tan  (a  ±  p)  ■=  ■ — — ^—^ 

I  =F  tan  a  tan  /3 

sin  a  +  sin  /3  =2  sin  ^  (a-J-/3)  cos^  (a  —  ^). 
sin  a  — sin  /3  =2  cos  J  (^4-/3)  sin  ^  (a  — /3). 
cosa  +  c«>sj3  =  2cos  J  (a4-/3)cos^  (a  — /3). 
cosa  — ct)S/3=— 2sin^(a  +  /3)sini(a  — /3). 


Fig. 


sin  a  sin  /3  =  ^  cos  (a  —  /3)  —  ^  cos  (a  +  ^) . 
cos  a  cos  /3  =r  J  cos  (a  —  j3)  +  ^  cos  (a  +  /3) . 
sin  a  cos  ^  =  ^  sin  (a  +  P)  +  ^  sin  (a  —  /3). 

sin2  a  -  sin2  /3  =  cos^jS  -  cos^a  =  sin  (a  +  /3)  sin  (a  -  jS). 
cos^  a  —  sin2  j3  =  cos'^  /3  —  sin^  a  =  cos  (a  +  /3)  cos  (a  —  /3). 


sin  2  a  =  2  sin  o  cos  a. 
cos  2  o  =  cos-  o  —  sin^  a. 

2  sin2  ^  a  =  I  —  cos  o 
tan 


tan  2  a  = 


2  tan  g 
I  -  tan2a 


i«=±Vr 


cos  a 


2  C0S2  ^  o  =   I   -f  cos  O. 

sin  a  I  —  COS  a 


+  COS  a       I  +  cos  a  sin  a 

sin  a  +  sin  (o  +  ;r)  +  sin  (a  -f  2  jc)  -f  •••  +  sin  (a  +  nx) 

_  sin  ^  («  4-  I )  jT  sin  (o  +  ^  nx) 
sin  i^x 

cos  a  f  cos  {a  +  x)+  cos  (a  +  2^)+  •••  +  cos  (a  -f  «jir) 

_sin^  («  +  0  .rcos(a  -f  ^  «^) 
~  sin  ^  jr 


■\/  —  I.  ^*  =  cos  X  +  «■  sin  jr. 

cos  X  =  -  (^»  -f  ^-**) . 

2 


^  •"  =  cosjT  —  t  smx. 
sin  jr  =  —  (<?«■  —  <»-**). 

2? 


^»«»  =  (cos  X  +  i  sin  x)"  =  cos  nx  + 


.f^  O-    THf 


% 


^l^^gSi 


or 


98 


PLANE   TRIANGLES. 


a  +  d  _tain^  (A  -\-  B) 

V^^b  ~  Xzxi\\A  -  B)  '^" 

a  —  h      _                  2  sin  ^  r    ,—r 
c  = ,  if  tan  X  = Vao. 


a     _      b      _      c 
sin  A      sin  B     sin  C  . 
a   =  <5cos  C+  ccosB. 

a^  =  IP-  +  c'  —  2  be  cos  A. 

a  sin  \{B  -  C)-(b-c)  cos  J  A. 
a  cos  ^(B-  C)  =  {b-\-  c)  sin  ^  A. 

asm  B 


c  —  a  cos  B 


\{  s=\{a  -\-  b  -^  c)\ 


smM  =  A/^— 4^ ^• 


/(^-^)(^-0 


.s^A=^jf^.        tanM  =  V^^^^ 

J(s-a)0-i)(s-^_     ,a„M=— •     tan  J  5=-^.     ta,>JC=-il-. 

'J  ^  s—a  ^  s  —  b  ^  s—c 

.  1     JL   •     ^      f2     sin  /4  sin  i5 

Area  —  \ab%yi\  C  =  —  ' 


2  sin  C 

Radius  of  inscribed  circle  =  r. 
Diameter  of  circumscribed  circle 


=  \/ s{s-  a)(^s-b)  {s-  c). 


sin  A 


DIFFERENTIAL  FORMULAS  FOR  PLANE  TRIANGLES. 


i/A  +  dB  +  dC  =  o. 

—  -  cot  A  dA  ='^  -  cotBc/B  =  -  -  cot  C  dC. 
a  b  c 


da  -  cos  Cdb  +  cos  B  dc  -{-  b  sin  CdA. 
a  dB  =  sin  Cdb  —  s'm  B  dc  —  b  cos  CdA. 


RIGHT   SPHERICAL  TRIANGLES  (C=90°). 


sin  «  =  sin  ^  sin  c. 
sin  a  =  cotB  ta.nb. 
a  cos  ^  =  sin  -5  cos  a. 
cos  A  =  tan  b  cot  c. 


sin   ^  =  sin  ^  sin  c. 
sin   <^  =  cot  .4  tana, 
cos  i9  =  sin  ^  cos  b. 
cos  B  =  tan  a  cot  r. 


cos  c  =  cos  a  cos  b  =  cot  -r4  cot  B. 


Fig.  3. 


99 


OBLIQUE   SPHERICAL  TRIANGLES. 


sin  a  _  sin^  _  sing 

sin  /^      sin  .5      sin  C 

cosrt  =      cos(^  cosr  +  sin  <^sinr  cos/^. 

cos  A  =—  cos  ^  cos  C  4-  sin  i9  sin  C  cos  a. 

sin  a  cos  i9  =  cos  b  sin  f  —  sin  b  cos  r  cos  /4. 

sin  A  cos  <J  =  cos  B  sin  C  +  sin  B  cos  C  cos  a. 

sin  a  cos  ^    =  sin  c  cos  ^  +  cos  a  sin  ^  cos  C. 
sin  ^  cos  B  =  cos  ^  sin  C  —  cos  c  cos  ^  sin  B. 
sin  rt  cot  /J    =  cot  i5  sin  C  -\-  cos  a  cos  C 
sin  ^  cot  B  =  cot  /^  sin  c  —  cos  r  cos  A. 


,      B 


s=\{a  +  b-\-c). 


sin2  1  v^ 
cos2  1  A  = 


sin  ^  sin  c 
sin  J  sin  (j  —  a) 


sin  /J  sin  c 

^  Sin  5  sin  (5  —  a) 

sin  (j  —  a)  sin  (j  —  b)  sin  (j  —  f) 


tan  ^  ^  = 


sin(j  —  a) 


tan^  \  a 


S=^(iA-\-B+  C). 
_  —  cos  S  cos  {S  —  A) 
sin  ^  sin  C 

_  cos  (5  -  B)  cos  (5  -  C) 
~  sin  ^  sin  C 

—  cos  5  cos  {S  —  A) 
co^(S-  B)cosiS-  C) 

—  cos  S 


cos  (5-^)  cos  (5  -  B)  cos  (5-  C) 


rt   =>?cos(5-^). 


sin  \c  s\n^{A  —  B)—  cos  ^  C  sin  i  («  —  b). 
sin  I  <r  cos  \'{A  —  B)=sm\  C  sin  ^  (a  +  ^). 
cos  I  f  sin  I  (y4  +  ^)  =  cos  |  C cos  ^  (a  —  /5). 
cos  \ c  cos  |(^  +  ^)  =  sin  \  Ccos l  («  +  ^). 


-  sin \{A-\-  B') 
tan  ^a  -  ^)  ""  sinT^^  -  B) 


tan  I , 


tan 


cos \(^A-^  B) 

tan  i  (a  +  b)  ~  cos  |  {A  —  B) 


cot\C        _sin^(g  +  ^) 
tan  i  (/4  -  B)  ~  sin  ^  {a  -  b) 

cot  ^  C        _  cos  \{a^-b) 
tan  |(/4  +  i9)  ~  cos  \{a-b>i 


r  =  tangent  of  the  angular  radius  of  the  inscribed  small  circle. 

R  =  tangent  of  the  angular  radius  of  the  circumscribed  small  circle. 


SPHERICAL  EXCESS. 


E  =  A  +  B-\-C-  i8o°. 

sm\a  sin^b   . 

s\nlt  E  = — ; —  sin  C. 

cos  ^  c 


tan^^ 


_       tan  ^rt  tan^^sin  C 
~  I  +  tan  ^  rt  tan  ^  b  cos  C 


tan2  J  ^  =  tan  ^  J  tan  ^  {s  -  a)  tan  ^  (j  -  <J)  tan  ^  (  j  -  0. 
jE"  =  area  -=-  r^sin  i". 


DIFFERENTIAL   FORMULAS   FOR   SPHERICAL 
TRIANGLES. 

cot  ada  —  cot  A  dA  =  cot  dd^  —  cot  B  dB  —  cot  cdc  —  cot  CdC. 

da  —  cos  Cdb  -\-  cos  B  dc  +  sin  c  sin  j5  a'/?. 

dA  =  sin  ^  sin  Cda  —  cos  r  ^Z?  —  cos  b  dC. 

sin  cdB  =  —  cos  <r  sin  B  da  -\-  sin  Adb  —  sin  /^  cos  ^  </C. 

MACLAURIN'S  THEOREM* 

/(^)=/(o)+/'(o)^+/'(o)^+/'"(o):^+.... 
I  2!  3! 

TAYLOR'S  THEOREM  * 
I  2!  3! 

dx  \       dy  \       dx^  2  !       ^^  2  !      dxdy 
where  u=f{x^  y), 

LAGRANGE'S  THEOREM.* 

«  =/"(2),  and  z=y  +  x(t>{z); 

BINOMIAL  THEOREM. 

I  I  •  2  I  •  2  •  3 

EXPONENTIAL  THEOREM.* 


M  \  M  I  2\      \  M  I  3\      \  A/  I  4\ 


^  =  I  +  ;c  +  -^2  +  7^  +  — ^  H ^  + ^  +  •-. 

2  6  24  120  720 


*  n  !  denotes  "  factorial  «,"  or  the  product  i  •  2  •  3  •  4  •••  «. 


LOGARITHMIC   SERIES.* 

,o.(...=,o....[.(i)--Q>;(^7-'(,7....|. 

log(i  +  x)=  A/  {x  -  ^  x-i  +  -X^  --Ji-*  +  -x^ ). 

23         4         5 

l0g(l    -  X)=:  -  M  {X  -^  ^~X^  -\-  ^  X^  +  ^X*  +  -  X^  -^  •••)• 


iVj//       2!^^//         3!V^'^/        4!\^/ 


+  -. 


TRIGONOMETRIC    SERIES.*  t 


I   3!   5!   7! 

;r'^  ,   x^       x^    , 

cos  ;r=l -\ — h""- 

2!   4!   6! 


3     15     315    2835     155925 
cot  X  = X x^ x^ 


•^   3    45    945    4725    93555 

sec^  =  I  +  i;c2  +  -^x^  +  ^x^  +  ^x^  +  .... 
2    24     720    8064 

cosec  ;r  =  -  +  -;i:+  -^ x^  -\ ^ x^-{- ^x''  +  •••. 

X      6    360     15120    604800 

sin-ij  =  v  + -y  + -^  vS  ^  _5_ y  +  _35_  y  +  .... 
6    40     112     1152 

-^2-^6-^  40-^  112-^  1152-^ 

tan-ijj/  =  J j3  4-  _y jj,7  _|_  _y  _  .... 

3  5  7  9 

>'    3r    5r    7r    9r 

log  sin  .r=  log  X  -  M  (  ^-x^  +  -^x^  -\-  -^x^  +  —^  x^  -i-  "\ 
V6  180  2835  37800  / 

\ogcos  X  =  -  A/  (  ~x^  -j-  —  x^  +  —x^  +  -^x»  +  "•]. 

\2  12  45  2520  / 

log  tan  or  =  log.r  +  Afl-x^  +  ^x^  +  -^x^  +  -^^  +  -V 
V3  90  2835  18900  I 

logsin-Jj  =  log;/  +  AI  f  iy^  +  -^-y  +  Jli  j^  +  ...). 
\.6  180  5070  / 

logtan-i/  =  log7-il/f-jj/2-i3y +  ^^/ V 

\3  90  2835  J 

logsin^  =  logtanjc-  Afl-tan'^x  -  -  tan*  ;ir  +  i  tan^  j;  --^tan^jr  +  ...  ). 
V2  4  6  8  / 

log  tan  X  =  log  sin  x  +  A/(  -  sin^  ;r  +  -  sin*  x  +  -  sin^  x  +  -  sin^  j:  +  ••  •  ) . 
\2  4  6  8  / 

*  n !  denotes  "  factorial  «,"  or  the  product  I  •  2  •  3  •  4  ...  «. 
t  The  angles  are  expressed  in  circular  measure. 


DIFFERENTIATION. 

d {ax  +  /J)  =  <z  dx.  d{ti  ±v)=  dii  ±  dv.  d {uv)  =  udv  +  v du. 


d 


dx 


(x\ydx-xdy^  ^(^")=«-r"-V^.  d  {Vx) 


2y/x 


d{\ogx)=M~-  d(a'')=~a^logadx.  d(e^)=^dx. 

d {xy)  =  xv  logejr  dy  +  yxv-'^  dx. 
d  {s\n  x)  =  cos  X  dx.  d{cosx)  =  — smxdx.  --^ 

d  (tan  jt)  =  sec^  x  dx.  d  (cot  x)  =  —  cosec^  x  dx. 

d  (sec  jt)  =  sec  x  tan  x  dx.  d(cosec  x)  =  —  cosec  x  cot  x  dx. 

d(sin-^x)=        '^-^      .  ^(tan-i^)=-i^.  ^(sec-i;r).-         '^"^ 


Vi  -x^                                     I  +  ^''  xV^^^^^ 

^^  v^^^*-i„A_         ^^  jf 1  ..N^  dx 


^(cos-i;r)  = '^^-.        ^(cot-ix)  =  --^,.        a'(cosec-i  x)  = - 


t/(vers-iAr)  =  — ^^=3.  ^(covers"!  ;r)  =  '^"^ 


y/zx  —  x"^  y/2x  —  x'^ 


APPROXIMATE  INTEGRATION. 

Let  Ajc  be  the  common  distance  between  the  ordinates  I'o,  ^1,  y^,  •••^n>  where 
«  is  even. 

J^  ^   ^  3  180  1512  ' 

where  P  =  Lx  [^'0  +  ;'„  +  2  {yo  +  _j/4  +  . . .  +  j„_2)  ] , 

^=:  2Ax[>'i  +  V3  +   •••   +J>'«-l], 

Fn'"  = f{x')   when  x  =  abscissa  of  jn* 

2.  Simpson's  rule : 

^  =  ^[JO  +  Jn  +  2  (^2  +^4    +•••  +JJ'„_2)+  4(>'l  +  J3  +   -  +/«-!)]. 

3 

3.  Weddle's  rule  (for  seven  ordinates)  : 

A  -  ^— ^ [jj'o  +  72  +  ^'4  +  jJ'6  +  jJ^3  +  5  (/I  4-/3+  J5)]. 
10 

4.  Prismoidal  formula :   F  =  —  [^  +  /4'  +  4  /^m]  =  -  [^  +  ^'  +  4  ^«]. 

3  6 


I03 


Constants. 
Base  of  Naperian  logs :  ^  =    .  . 

Modulus  of  common  logs :  log  e  •=.  M  ■■ 
Degrees  in  arc  =  radius:    180'^  h-  tt  = 

Minutes  in  arc  =  radius : 

Seconds  in  arc  =  radius  : 

360^^  expressed  in  minutes  of  arc  :  .  .  . 
360°  expressed  in  seconds  of  arc  :  .  .  . 
24  hours  expressed  in  minutes  of  time : 
24  hours  expressed  in  seconds  of  time: 
TT  =  314159  26535  89793  23846 
logTT  =0.49714  98726  94133  85435 
sin  i"  =0.00000  48481  3681 1  07637 
arc  i"  =0.00000  48481   3681 1  09536 


.  .  .  .    2.71828  183    . 
.  .  .  .    0.43429448    . 

•  •  .  57°-29577  95i    • 

•  .3  437'-74677-  •  •  • 
2o6  264".8o6     .  .  .  . 

21  600'    . 

I  296000"  . 

I  440"*  . 

86400*    . 


Eng.  inch 0.02540 

Eng.  foot 0.30480 

Eng.  yard 0.91440 

Eng.  statute  mile 1.60935 

meter 39-3700 

meter 3.28083 

meter i. 09361 

kilometer O.62137 

sq.  foot 9.29034 

sq.  inch 6.45163 

sq.  meter 10.7639 

sq.  centimeter 0.15500 

cubic  foot 0.02831 

cubic  inch     16.3872 

cubic  meter 35-3145 

cubic  decimeter  (liter) .  .  .  61.0234 

avoirdupois  pound 453.59242 

avoirdupois  ounce 28.34953 

Troy  ounce 31.10348 

grain 64.79892 

kilogram 2.20462 

kilogram 35.2740 

kilogram 32.1507 

gram 15-43235 

foot-pound 0.13825 

kilogram-meter 7.23300 

pound  per  sq.  in 70.3067 

gram  per  sq.  cm 0.01422 

pound  per  cu.  ft 0.0 1 601 

grain  per  cu.  in 0.00395 

gram  per  cu.  cm 62.4283 

gram  per  cu.  cm 252.8925 


70 


77 


639 
5 


meters 

meters 

meters 

kilometers  .... 
Eng.  inches   .  .  . 

Eng.  feet 

Eng.  yards  .... 
Eng.  statute  miles 

sq.  decimeters  .  . 
sq.  centimeters    . 

sq.  feet 

sq. inches    .... 

cubic  meters  .  .  . 
cubic  centimeters 
cubic  feet  .... 
cubic  inches  .  .  . 


grams  .  .  .  . 
grams  .  .  .  . 
grams  .  .  .  . 
milligrams  .  . 
avdp.  pounds 
avdp.  ounces 
Troy  ounces  . 
grains    .  .  .  . 


kilogram-meters 
foot-pounds   .  . 


grams  per  sq.  cm. 
34  lbs.  per  sq.  in.  .  . 
84  grams  per  cu.  cm. 
425    grams  per  cu.  cm. 

lbs.  per  cu.  ft.   .  . 

grains  per  cu.  in. 


Logarithms. 
0.43429  448 

9.6377^  431 
1. 75812  263 

3.53627  38« 
5.31442  513 

4-33445  375 
6. 1 1 260  500 
3.15836249 

4-93651  374 
0.49714987 


.68557487  —  10 
-68557  487  -  10 

Logarithms. 
8.40483  5  —  10 
9.48401  6  —  10 
9.961137-  10 
0.20665  O 

I-595165 
0.515984 
0.03886  3 
9-79335  o  -  10 
0.96803  2 
0.80966  9 
1.031968 
9.19033  I  -  10 

8.45204  7—10 
1.214504 

1-54795  3 
1.785496 

2.65666  6 
1.452546 
1 .49280  9 
1.811568 

0-34333  4 
1.54745  4 
1.50719  I 
1. 18843  2 

9.14068  2—10 
0.85931  8 
1.84699  7 
8.153003  -  10 
8.20461  8  —  10 
7.597064-  lO- 
1.795382 
2.40293  6 


Logarithms. 
dynes  (^jf  in  meters). 

dynes  (^  in  meters) 0.811568 

ergs  {g'\v\  centimeters)  ....  4.140682 
ergs  (^  in  centimeters), 
ergs  per  sec. 

watts  (^  in  meters) 1.881044 

watts  (approximately)   ....  2.87273  9 

32.086  528  +  0.171  293  sin2  0  —  0.000003  h.  in  feet  (Harkness). 
=    9-779  886  +  0.052  210  sin^t^  —  0.000003  f^-  i"  meters  (Harkness). 
/  =  39.012  540  -f  0.208  268  sin2  0  —  0.000000  3  //.  in  inches  (Harkness) 
=    0.990  910  -f  0.005  290  sin2  0  —  0.000  000  3  h.  in  meters  (Harkness). 


Wt.  of  mass  of  1  gram  .  .  loo^ 

Wt.  of  mass  of  I  grain  .  .  6.47989  2g 

1  foot-pound 1 3825.5  i" 

1  kilogram-meter looooo^ 

I  watt lo'' 

I  horse-power 76.0404^ 

I  horse-power 746 


g 


EXPLANATION    OF   THE    TABLES. 
INTRODUCTORY. 

1.  When  we  have  a  number  with  six  or  more  decimal  places,  and  we 
wish  to  use  only  five  : 

{a)  If  the  sixth  and  following  figures  of  the  decimal  are  less  than 
5  in  the  sixth  place,  they  are  dropped  ;  thus,  0.46437  4999  is  called 
0.46437. 

(^)  If  the  sixth  and  following  figures  of  the  decimal  are  greater  than 
5  in  the  sixth  place,  the  fifth  place  is  increased  by  unity  and  the  sixth 
and  following  places  are  dropped  ;  thus,  0.46437  5001  is  called  0.46438. 

(c)  If  the  sixth  figure  of  the  decimal  is  5,  and  if  it  is  followed  only  by 
zeros,  make  the  fifth  figure  the  nearest  even  figure ;  thus,  0.46437  500  is 
called  0.46438,  while  0.46438  500  is  also  called  0.46438.  The  number 
is  thus  increased  when  the  fifth  figure  is  odd,  and  decreased  when  it  is 
even,  the  two  operations  tending  to  neutralize  each  other  in  a  series  of 
computations,  and  hence  to  diminish  the  resultant  error. 

2.  Hence  any  number  obtained  according  to  Art.  i  may  be  in  error 
by  half  a  unit  in  the  fifth  decimal  place. 

3.  When  the  last  figure  of  a  number  in  these  tables  is  5,  the  number 
printed  is  too  large,  the  5  having  been  obtained  according  to  Art.  i  (^) ; 
if  the  5  is  without  the  minus  sign,  the  number  printed  is  too  small, 
the  figures  following  the  5  having  been  dropped  according  to  Art.  i  (a). 

4.  The  marginal  tables  contain  the  products  of  the  numbers  at  the 
top  of  the  columns  by  i,  2,  3,  •••9  tenths,  and  may  be  used  in  multiply- 
ing and  dividing  in  interpolation. 

{a)   To  multiply  38  by  .746  : 

38  X  .7  =  =  26.6 

38  X  .4=  15-2;  •••  38  X  .04    =    1.52 
38  X  .6  =  22.8  ;  .-.  38  X  .006  =      .228 
.-.  38  X  .746  =  28.348 


In  multiplying  by  the  second  figure  (hundredths),  the  decimal  point 
in  the  table  is  moved  one  place  to  the  left ;  in  multiplying  by  the  third 
(thousandths),  two  to  the  left ;  and  so  on. 


88 

I 

3-8 

2 

7.6 

3 

1 1.4 

4 

15.2 

s 

19.0 

6 

22.8 

7 

26.6 

8 

30.4 

9 

34.2 

(105) 


EXPLANATION   OF   THE   TABLES. 


{b)  To  divide  28  by  t,^  : 


Dividend, 

28 

38 

Next  less, 

26.6 

corresponding  to 

•7 

I 

2 

3.8 
76 

Remainder, 

I  4 

3 

11.4 

Next  less, 

I  1.4 

corresponding  to 

•03 

4 

5 

15.2 
19.0 

Remainder, 

26 

6 
7 

22.8 
26.6 

Nearest, 
.  Quotient, 

26.6 

corresponding  to 

.007 
.737 

8 
9 

304 
34-2 

to  the  nearest  third  decimal  place.     The  decimal  point  is  moved  one 
place  to  the  right  in  each  remainder,  since  the  next  figure  in  the  quotient 
will  be  one  place  farther  to  the  right. 
To  divide  23  by  38  : 

Dividend,      23 

22.8         corresponding  to  .6 


0.0      corresponding  to  .00 


2  o. 


Nearest, 
.-.  Quotient, 


I  9.0  corresponding  to  .005 
.605 


The  computer  should  use  the  marginal  tables  mentally. 


LOGARITHMS. 


5.  The  logarithm  of  a  number  is  the  exponent  of  the  power  to  which 
a  given  number  called  the  dase  must  be  raised  to  produce  the  first 
number.  U  A  =  <?",  a  is  called  the  logarithm  of  the  number  A  to  the 
base  <r,  written  log,  A  =  a. 

6.  If  A  =  e'',  and  B  =  e'',  or  (omitting  subscripts)  log  A  =  a,  and 
log  B  =  d,  wc  have 

AxB  =  e''+';  .'.  \og(A  X  B)  =  a -\- d  ;  .'.  \og{A  xB)=\og  A-\-\ogB. 
A-^-B^e--"',  .'.  \og{A^B)  =  a-b;  .\  \og{A  ^  B)  =\og  A -log  B . 
^»         rzz^"-;     .-.  log(^")         =na',        /.  log(^")        =n\ogA. 


V^      =e"';     .'.  logV^  =-a; 


.-.  logv^        =-\ogA. 
It 


EXPLANATION   OF   THE   TABLES.  iii 

7.  When  the  base  is  not  specified,  it  is  generally  understood  that 
logarithms  to  the  base  lo,  or  common  logaritfwis y  are  meant.  In  this 
system,  since 


o.ooi 

= 

T 
lOOG 

I 

~IO» 

= 

^o-^ 

log  o.ooi  =-3; 

CO  I 

= 

I 
lOO 

I 
~  lo- 

= 

io-\ 

log  o.oi    =  —  2  ; 

O.I 

= 

I 

lO 

I 

~  lO 

= 

lO-S 

log  0.1      =-i; 

I. 

^ 

IO^ 

log        1=0; 

lO. 

= 

io\ 

log      10  =  +  I  ; 

lOO. 

= 

lO^ 

log    100  =  +  2; 

lOOO. 

= 

lo^ 

log  1000  =  4-3. 

8.  The  logarithm  of  a  number  between  100  and  1000  will  be  a  num- 
ber between  2  and  3,  or  2  +  ^  where  m  will  be  a  decimal  called  the 
mantissa^  the  integral  portion  of  the  logarithm  being  the  characteristic. 
The  mantissa  is  always  considered  positive;  thus  log 0.002  will  be  a 
number  between  —  2  and  —  3,  that  is,  either  — 34-w  or  —  2  —  m\ 
the  first  form  being  used.  We  write  log  0.002  =  3-30103,  the  negative 
sign  being  placed  over  the  characteristic  to  show  that  the  characteristic 
alone  is  negative. 

9.  Since 

log  {A  X  10**)  =  log  A  +  log  10"  =  log  A  ■\-  n  log  10  =  log  A  -{-  rij 
and   log  (A  --;  10")  =  log  A  —  log  10"  =  log  A  —  n  log  10  =  log-^  —  n, 

we  have,  if  log  37.3=  1.57171, 

log373-  =2.57i7i»  and  log 3.73  =0.57171, 
log3730  =3-57171,  and  log 0.373  =i-57i7i; 
log 37300  =  4-5 71 71,   and   log 0.0373  =  2.57171. 

Hence  the  position  of  the  decimal  point  affects  the  characteristic  alone, 
the  mantissa  being  always  the  same  for  the  same  sequence  of  figures. 
For  this  reason  the  common  system  of  logarithms  is  used  in  practice. 

10.  The  characteristic  is  found  as  follows :  When  the  number  is 
greater  than  i,  the  characteristic  is  positive ^  and  is  one  less  than  the  num- 
ber of  digits  to  the  left  of  the  decimal  point ;  when  the  number  is  less 
than  I,  the  characteristic  is  negative,  and  is  one  more  than  the  number 
of  zeros  between  the  decimal  point  and  the  first^signij^cantfigj^re. 

11.  To  avoid  the  use  of  negative  characteristics  we  add  10  to  the 
characteristic  and  write  —10  after  the  mantissa,  i.e.  adding  and  subtract- 
ing the  same  quantity,  10.     Thus  log  0.2  =  T. 30103  would  be  written 


iv  EXPLANATION  OF  THE  TABLES. 

9.30103  —  10.  The  —  10  is  often  omitted  for  brevity  when  there  is  no 
danger  of  confusion,  but  its  existence  must  not  be  forgotten.  Such 
logarithms  are  called  augmented  logarithms. 

Jfi  this  case  the  chai-acteristic  of  the  logarithfn  of  a  pure  decwial  is  9 
diminished  by  the  niiniber  of  ciphers  to  the  left  of  the  fi?'st  significant 
figure.  Thus  the  characteristic  of  log  0.004  is  9  —  2,  or  7,  and  that  of 
log  0.94  is  9  —  o,  or  9. 

12.  The  arithmetical  complement  of  the  logarithm  (written  co/og) 
of  a  number  is  the  logarithm  of  its  reciprocal,  and  is  found  by  subtract- 
ing each  figure  of  the  logarithm  from  9,  commencing  at  the  left,  except 
the   last   significant  figure  on  the  right,  which  is  subtracted  from   10. 

For  \og  —  =  —  \ogx=io  —  \ogx—io; 

thus,  if  log^=  2.46403,  colog^=  7.53597- 10; 

if  log  .a;  =  8.43000  —  10,  colog.T  =  1.57000. 

TABLE    L 

13.  Page  3  contains  the  logarithms  of  numbers  from  i  to  100,  to 
five  decimal  places. 

Pages  4-21  contain  the  mantissas  of  the  logarithms  of  numbers  from 
1000  to  10009,  to  five  decimal  places. 

Pages  22,  23,  contain  the  mantissas  of  the  logarithms  of  numbers 
from  1 0000  to  1 1009,  to  seven  decimal  places. 

NoTt;.  — The  mantissas  of  the  logarithms  of  numbers,  except  those  of  the  integral 
powers  of  10,  are  incommensurable,  the  mantissas  in  the  tables  being  found  as 
shown  in  Art.  i. 

To  find  the  Logarithfn  of  a  Number. 

14.  The  characteristic  is  found  by  the  rules  in  Arts.  10  and  11,  and 
the  77iantissa  from  the  tables,  as  shown  in  Arts.  15,  16,  17,  18. 

15.  Wheji  the  number  has  four  figures.  —  Find  on  pages  4-21  the 
first  three  figures  in  the  column  marked  N,  and  the  fourth  at  the  top 
of  one  of  the  other  columns.  The  last  three  figures  of  the  mantissa  are 
found  in  this  column  on  the  horizontal  line  through  the  first  three 
figures  of  the  given  number  in  column  N.  The  first  two  figures  of  the 
mantissa  are  those  under  L  in  the  same  line  with  the  number,  or  else 
those  nearest  above  it,  unless  the  last  three  figures  of  the  mantissa  as 
given  in  the  tables  are  preceded  by  a  *,  when  the  first  two  figures  are 
found  under  L  in  the  first  line  below  the  number.     Thus  (page  4), 

log  1136  =  3.05538;  log  1137  =  3.05576;  log  1 138  =  3.05614; 

log  1370  =  3.13672 ;  log  1371  =  313704 ;  ^og  1372  =  3-13735 ; 
log  1380  =  3-13988 ;  log  1381  =  3.14019  ;  log  1382  =  3-14051- 


EXPLANATION   OF   THE   TABLES.  V 

16.  When  the  nuviber  has  less  than  four  figures,  annex  ciphers  on 
the  right  and  proceed  as  in  Art.   15.     Thus, 

log  1. 13  =  0.05308;  log  12.8=  1.10721  ;  log  130=  2.1 1394;  J 

log  15      =  I.I 7609;  log  16     =  1. 20412  ;  log  17    =1.23045.        / 

17.  When  the  nutnber  has  more  than  fotir  figures,  as  11.4672. — 
Since  the  mantissa  is  independent  of  the  position  of  the  decimal  point, 
point  off  the  first  four  figures  and  find  the  mantissa  of  log  1146.72. 
This  will  be  between  the  mantissas  of  log  1 146  and  log  1 147.  Hence 
find  from  the  tables  the  mantissas  corresponding  to  1146  and  1147; 
multiply  the  difference  between  them  (called  the  tabular  difference)  by 
.72,  and  add  the  product  (called  the  correction)  to  log  11.46;  the 
result  will  be  the  logarithm  required. 

Mantissa  of  log  1 146  =  05918  ,      log  11.46  =  1.05918 

Mantissa  of  log  1147  =05956  correction' ==  38  x  .72  =  27.36 

Tabular  difference      =        38  .-.  log  11.4672  =  1.05945  36 

or  =  1.05945 

Note.  — Since  any  mantissa  in  the  tables  may  be  in  error  by  half  a  unit  in  the 
fifth  decimal  place  (Art.  2),  no  advantage  is  gained  by  using  the  sixth  place  in 
the  interpolated  logarithm.  Thus,  according  to  Art.  i,  we  drop  the  .36,  and  call 
log  11.4672=  1.05945. 

Note.  — The  marginal  tables  should  be  used  in  multiplying  the  tabular  difference 
to  find  the  correction  (Art.  4). 

Note.  —  It  is  assumed  that  the  change  in  the  mantissa  is  proportional  to  that  in 
the  number,  as  the  latter  increases  from  1146  to  1147.  An  increase  of  i  in  the  num- 
ber causes  an  increase  of  38  in  the  mantissa;  hence  an  increase  of  .72  in  the  number 
will  cause  an  increase  of  38  x  .72  in  the  mantissa. 

Note.  — We  could  also  find  the  mantissa  of  log  11.4672  by  subtracting  the  prod- 
uct of  the  tabular  difference  by  .28  (or  i.oo  —  .72)  from  the  mantissa  corresponding 
to  1147;  that  is,  the  required  mantissa  is  05956— (38  X  .28)  =05956— 10.64=05945 
as  before. 

18.  The  general  rule  is :  Find  the  fnantissa  corresponding  to  the 
first  four  figures  of  the  number ;  multiply  the  tabular  difference  by  the 
fifth  and  following  figures  treated  as  a  decimal;  and  add  the  product 

to  the  mantissa  just  found. 

The  tabular  difference  is  the  difference  between  the  mantissas  corre- 
sponding to  the  two  numbers  in  the  tables,  between  which  the  given 
number  lies. 

log  1.62163  =  0.20995  ;  logo.38o24  =  T.58oo6;  log 0.085 2  763  =  2.93083  ; 
log  189.524=  2.27767;  logo.386o2=T.5866i  ;  log 0.0085238  =  3.93419  ; 
log  19983.4  =  4.30067  ;  log3.98743  =  o.6oo7o  ;  logo.090046   =2.95446. 


vi  EXPLANATION   OF   THE   TABLES. 

Note.  —  Page  3  is  used  when  the  number  contains  less  than  three  figures,  the 
number  being  found  in  the  column  N',  and  the  logarithm  in  the  column  headed  Log. 
The  characteristic  is  given  for  whole  numbers,  and  must  be  changed  for  decimals. 

Note. — When  a  number  is  composed  of  three  figures,  find  on  pages  4-21  the 
number  in  the  column  A',  and  the  mantissa  corresponding  in  the  column  L.  o. 

To  find  thf  Number  co7-responding  to  a  Giveii  Loga7'ithm. 

19.  From  the  tables  we  find  the  sequence  of  figures  corresponding 
to  the  given  mantissa,  as  shown  in  Arts.  20,  21,  and  22,  the  position  of 
the  decimal  point  being  determined  by  the  characteristic  (Arts.  10,  11). 

20.  When  the  given  mantissa  can  he  found  in  the  tables.  —  Find  on 
pages  4-2 1  the  first  two  figures  of  the  mantissa  under  L  in  "the  column 
headed  Z.  o.  The  last  three  figures  of  the  mantissa  are  then  sought  for 
in  the  columns  headed  o,  1,  2,  •••  9,  in  the  same  line  with  the  first  two 
figures,  or  in  one  of  the  lines  just  below,  or  in  the  line  next  above 
(where  they  would  be  preceded  by  a  *).  The  first  three  figures  of  the 
required  number  will  be  found  in  the  column  headed  N,  in  the  same 
horizontal  line  with  the  last  three  figures  of  the  mantissa,  and  the  fourth 
figure  of  the  number  at  the  top  of  the  column  in  which  the  last  three 
figures  of  the  mantissa  are  found.     Thus  (page  4), 

.    0.06221  =  log  1.154  ;  0.06558  —  log  1. 163  ;  0.06893  =  log  1. 172  ; 
0.07004  =  log  1. 175  ;  0.07188  =  log  1. 180  ;  0.08063  =  log  1.204. 

21.  When  the  given  mantissa  can  not  be  found  in  the  tables.  — If  we 
wish  to  find  the  number  whose  logarithm  is  2.1 6531,  we  enter  the  tables 
with  16531,  and  find  that  it  lies  between  16524  and  16554,  so  that  the 
given  mantissa  corresponds  to  a  number  between  1463  and  1464.  Also 
1 65 3 1  exceeds  16524  by  7,  and  this  difference,  divided  by  the  tabular 
difference  30,  gives  .23"'  as  the  amount  by  which  the  required  number 
exceeds  1463.  Hence  2.16531  =  log  146.323  •••,  which  we  call  146.32, 
according  to  Art.  i,  the  incompleteness  of  the  tables  making  the  sixth 
figure  uncertain. 

Note. — The  marginal  tables  should  be  used  in  dividing  the  difterence  between 
the  given  mantissa  and  the  one  next  less  in  the  tables  by  the  tabular  difference. 

22.  The  general  rule  is:  Find  the  7iufnber  corresponding  to  the 
mantissa  in  the  tables  next  less  than  the  given  ?najitissa  ;  divide  the 
excess  of  the  given  mantissa  over  the  one  found  in  the  tables  by  the 
tabular  dijference ;  and  annex  the  quotient  to  the  first  four  figures 
already  found. 

The  tabular  difference  is  the  difference  between  the  two  mantissas  in 
the  tables,  between  which  the  given  mantissa  lies. 

1.16600  =  log  g. 14656  ;    0.18002  =  log  1. 5136  ;    2.18200  =  log  152.06  ; 
1. 19000  =  log  15.488  ;      4.19680  =  log  15773  ;      1.20020  =  log  15.856. 

23.  For  the  use  of  the  numbers  S\  T\  S",  T",  see  Arts.  35-38. 


EXPLANATION   OP^  THE   TABLES.  vii 


TABLE    IL 

24.  This  table  (pages  26-70)  contains  the  logarithms,  to  five  deci- 
mal places,  of  the  trigonometric  sines,  cosines,  tangents,  and  cotangents 
of  angles  from  0°  to  90°,  for  each  minute.  The  logarithms  in  the 
columns  headed  Z.  6/;/,  Z.  Tan,  and  Z.  Cos,  are  augmented,  and  should 
be  diminished  by  10  (Art.  11),  while  those  in  the  columns  headed 
Z.  Cot  are  correctly  given. 

25.  Since  sec:r  = ,  and  cosec;c  =  -^ — ,  the  logarithms  of  the 

co'ix  sm^ 

secant  and  cosecant  of  an  angle  are  the  arithmetical  complements  of 
those  of  the  cosine  and  sine  respectively  (Art.  12). 

To  find  the  Logarithmic  Functions  of  an  Angle  Less  than  90°. 

26.  When  the  angle  is  less  than  45°,  the  degrees  are  found  at  the  top 
of  the  page,  and  the  minutes  on  the  left.  The  numbers  in  the  same 
horizontal  line  with  the  minutes  of  the  angle  are  the  logarithmic  functions 
indicated  by  the  notation  at  the  top  of  the  columns.      Thus  (page  34), 

log  sin  8°  4'  =  9.14714  —  10,         log  tan  8°  4/  =  9.15 145  —  10, 
log  cot  8°  4'  =  0.84855,  log  cos  8°  4'  =  9.99568  —  10. 

27.  When  the  angle  is  greater  than  45°,  the  degrees  are  found  at  the 
botto7n  of  the  page,  and  the  minutes  on  the  7'ight.  The  numbers  in  the 
same  horizontal  line  with  the  minutes  of  the  angle  are  the  logarithmic 
functions  indicated  by  the  notation  at  the  bottom  of  the  columns.  Thus 
(page  34), 

log  sin  81°  25'  =  9.995 1 1  —  10,     log  tan  81°  25'  =  0.82120, 

log  cot  81°  25'  =  9.17880  —  10,     log  cos  81°  25'  =  9.17391  —  10. 

28.  When  the  angle  is  given  to  decimals  of  a  minute.  —  In  finding 
log  sin  30°  8'.48,  for  example,  we  see  that  it  will  lie  between  the 
logarithmic  sines  of  30°  8'  and  30°  9',  that  is,  between  9.70072  and 
9.70093,  their  difference  21  being  the  change  in  the  logarithmic  sine 
caused  by  a  change  of  i'  in  the  angle.  Hence,  to  find  the  correction 
to  log  sin  30°  8'  that  would  give  log  sin  30°  8'.48  we  multiply  21  by  .48 
(Art.  4).  The  product  10.08  added  to  log  sin  30°  8',  since  log  sin  30°  9' 
is  greater  than  log  sin  30°  8"^,  gives  log  sin  30°  8 '.48  =  9vZoo82  (Art.  i). 
Similarly,  log  tan  30°  8'.48  =  9.76391,  log  cot  30°  8'.48  =  0.23609,  log 
cos  30°  8'.48  =  9.93691,  the  correction  being  subtracted  in  the  last 
two  cases,  since  the  cotangent  and  the  cosine  decrease  as  the  angle 
increases. 


viii  EXPLANATION   OF   THE   TABLES. 

29.  The  general  rule  is :  Fijid  the  function  corresponding  to  the  given 
degrees  and  minutes ;  tnultiply  the  tabular  difference  by  the  decimals  of 
a  minute;  add  the  product  to  the  function  corresponding  to  the  given 
degrees  and  minutes  when  finding  the  logarithmic  sine  or  tangent,  and 
subtract  it  ivhen  finding  the  logarithmic  cosine  or  cotangent. 

The  tabular  differences  are  given  in  the  columns  headed  d.  and  c.  d., 
the  latter  containing  the  common  difference  for  the  L.  Tan  and  Z.  Cot 
columns.  The  difference  to  be  used  is  that  between  the  functions  cor- 
responding to  the  two  angles  between  which  the  given  angle  lies. 

For  3o°39'.38:  log  sin  =  9.70747  ;   log  cos  =  9.93462  ; 

log  tan  =  9.77285  ;  log  cot  =  0.22715. 
For  59°  43 '.46:  log  sin  =  9.93632;  log  cos  =  9.70257  ; 

log  tan  =  0.23375  ;  log  cot  =  9.76625. 

30.  When  the  angle  is  given  to  seconds,  the  correction  may  be  found 
by  multiplying  the  tabular  difference  by  the  number  of  seconds,  and 
dividing  the  product  by  60. 

To  find  the  Acute  Angle  corresponding  to  a  Givefi  Logarithmic 
Function. 

31.  The  column  headed  L.  Sin  is  marked  L.  Cos  at  the  bottom,  while 
that  headed  Z.  Cos  is  marked  Z.  Sin  at  the  bottom  ;  hence,  if  a  logarith- 
mic sine  or  cosine  were  given,  we  should  expect  to  find  it  in  one  of  these 
two  columns.  Similarly,  we  should  expect  to  find  a  given  logarithmic 
tangent  or  cotangent  in  one  of  the  two  columns  headed  Z.  Tan  and 
Z.  Cot. 

32.  When  the  function  can  be  found  in  the  tables.  —  If  a  logarithmic 
sine  is  given,  find  it  in  one  of  the  two  columns  marked  Z.  Sin  and 
Z.  Cos ;  if  found  in  the  column  headed  Z.  Sin,  the  degrees  are  taken 
from  the  top,  and  the  minutes  from  the  left  of  the  page ;  if  in  the 
column  headed  Z.  Cos  but  marked  Z.  Sin  at  the  bottom,  the  degrees 
are  taken  from  the  bottom,  and  the  minutes  from  the  right  of  the  page. 
Thus, 

9.70115  =  log  sin  30°  10';  9.93457  =  log  sin  59^20'; 

9.93724  =  log  cos  30°    4';  9.70590  =  log  cos  59°  28'; 

9.76406  =  log  tan  30°    9';  0.23130  =  log  tan  59°  35'; 

0.23420  =  log  cot  30°  15';  9.76870  =  log  cot  59°  35'. 

33.  When  the  function  can  not  be  found  in  the  tables.  —  If  we  wish  to 
find  the  angle  whose  logarithmic  sine  is  9.70170,  we  see  on  page  56 
that  the  given  logarithmic  sine  lies  between  9.70159  and  9.70180,  and 


'     .^    .       .^EXPLANATION   OP^   THE   TABLES.       *  ix 

hence  the  angle  is  between  30°  12'  and  30°  13'.  The  given  logarithmic 
sine  differs  from  log  sin  30°  12'  by  11,  and  this  difference,  divided  by 
the  tabular  difference  21,  gives  .52+  as  the  decimal  of  a  minute  by 
which  the  angle  exceeds  30°  12'.  Hence  9.70170  =  log  sin  30°  i2'.52, 
wliich  we  call  30°  i2'.5,  since  the  incompleteness  of  the  tables  (Art.  i) 
makes  the  hundredths  of  a  minute  uncertain. 

34.  T/ie  rule  is :  For  a  logarithmic  sifie  or  tangent  find  the  degrees 
and  minutes  corresponding  to  the  function  in  the  tables  7iext  less  thati 
the  given  function  ;  divide  the  differeiice  between  the  given  function  and 
the  ofte  next  less  by  the  tabular  difference ;  and  the  quotient  will  be  the 
decimal  of  a  minute  to  be  added  to  the  degrees  and  minutes  already 
foujid.  For  a  loga7-ithmic  cosine  or  cotangent  find  the  degrees  and  min- 
utes cor7'esponding  to  the  function  next  greater  than  the  given  function, 
sitice  the  cosine  and  cotangent  decrease  as  the  angle  increases,  and  divide 
the  difference  between  the  given  function  and  the  one  next  greater  by  the 
tabular  diffeirnce,  to  find  the  decimal  of  a  minute. 

The  tabular  difference  is  the  difference  between  the  two  functions  in 
the  tables,  between  which  the  given  function  lies. 

9.70000  =  log  sin  30°  4'.7;  9.93500  =  log  sin  59°  25'.7  ; 
9.93400  =  log  cos  30°  47'.6  ;  9.70500  =  log  cos  59°  32'.2  ; 
9.77000  =  log  tan  30°  29 '.5  ;  0.23200  =  log  tan  59°  37'.4  ; 
0.23300  =  log  cot  30°  19'.!  ;    9.76400  =  log  cot  59°  5 1 '.2. 

Angles  Near  0°  or  90°. 

35.  The  assumption  that  the  variations  in  the  functions'  are  propor- 
tional to  the  variations  in  the  angles  if  the  latter  are  less  than  i '  fails 
when  the  angle  is  small,  shown  by  the  rapid  changes  in  the  tabular 
differences  on  pages  26,  27,  and  28. 

36.  The  quantities  S'  and  7"' which  are  used  in  this  case  are  defined 
by  the  equations 

Of      1      sin  a 

a' 

rr>i      1       tan  a 
^  =log — T' 
a' 

where  «'  is  the  number  of  minutes  in  the  angle.  Their  values  from 
0°  to  1°  40'  (=100')  are  given  at  the  bottom  of  pages  3-21  ;  from 
i°4o'to  3°  20'  at  the  left  margin  of  pages  4  and  5,  the  first  three 
figures  being  found  at  the  top ;  and  from  3°  to  5°  on  page  24.     Thus, 

for  i'=      i'   (pages),     ^' =  6.46  373,   ^' =  6.46  373  ; 

for  15'=:    15'  (pages),     ^' =  6.46  372,   r' =  6.46  373  ; 

for     2° 40'=  160'  (pages),     ^'=6.46357,   r'=  6.46404; 

for     4°  20' =260'  (page  24),  6*' =  6.46  331,  7"'  =  6.46  456. 
Each  of  these  numbers  should  have  —10  written  after  it  (Art.  11). 


X  EXPLANATION   OF   THE   TABLES. 

Note.  — The  logarithmic  cosine  of  a  small  angle  is  found  by  the  ordinary  method. 
The  cotangent  of  an  angle  is  the  reciprocal  of  the  tangent,  and  hence  the  logarithmic 
cotangent  is  the  arithmetical  complement  of  the  logarithmic  tangent.  The  formulas 
for  finding  the  logarithmic  cosine,  tangent,  and  cotangent  of  angles  near  90°  are 
given  on  page  25. 

37.  To  find  the  logarithmic  sine  or  tangent  of  a  small  angle.  —  From 
Art.  36,  we  have 

log  sin  «  =  6"'  +  log  «', 
log  tan  «  =  7"'  +  log  «'. 

Hence,  to  find  the  logarithmic  sine  or  tangent  of  an  angle  less  than 
5°,  find  the  value  of  the  6"'  or  T^  corresponding  to  the  angle,  interpolat- 
ing if  necessary,  and  add  it  to  the  logarithm  of  the  number  of  minutes 
in  the  angle. 

Find  log  sin  o°42'.6.     Since  the  angle  is  nearer  43'  than  42',  we  take 

6"' =6.46  371 
log  42.6  =  1.62  941 

.*.  log  sin  0°  42 '.6  =  8.09  312 

Find  log  tan  i°53'.2.     Since  the  angle  is  nearer  i°53'  (=  113')  than 

114',  we  take 

r'  =  6.46  388 
log  113.2  =  2.05  385 

/.  log  tani°53'.2  =  8.51  773 

Note.  —  When  the  angle  is  given  in  seconds,  either  reduce  the  seconds  to  deci- 
mals of  a  minute,  or  use  the  values  of  6""  and  7^"  given  at  the  bottom  of  pages 
3-23  and  on  page  24.     They  are  defined  by  the  equations 

5"=log^-i^,  and   7^"  =  log^, 
a"  a" 

where  a"  is  the  number  of  seconds  in  the  angle.     Hence 

log  sin  a  =  .S"  +  log  a",  and  log  tan  a—  T"  -\-  log  a". 

38.  To  find  the  S7?iall  angle  corresponding  to  a  given  logarithmic  sine 
or  tangent.  —  From  Art.  36, 

log  «'  =  log  sin  «  —  .S',  ' 

log  a'  =  log  tan  a  —  T\  . 

or  log  a'  =  log  sin  a  +  cpl  S ', 

log  «'  =  log  tan  a  +  cpl  T\ 

When  the  angle  is  less  than  3°,  find  on  pages  26-28  the  value  of 
cpl  6"'  (or  cpl  7")  corresponding  to  the  function,  interpolating  if  neces- 
sary, and  add  it  to  log  sin  «  (or  log  tan  «)  ;  the  sum  will  be  the  loga- 
rithm of  the  number  of  minutes  in  the  angle. 

In  finding  the  angle  whose  logarithmic  sine  is  8.09006,  we  see  from 


EXPLANATION   OF   THE   TABLES. 


XI 


theZ.  Sin  column  (page  26)  that  the  angle  is  between  0°  42'  and  o°43', 

and  that  the  value  of  cpl  S*  must  be  either  3.53628  or  3.53629.     The 

given  logarithmic  sine  is  nearer  that  of  42'  than  that  of  43';  hence  we 

take 

cpl  ^'  =  3-53628 

log  sin  a  =  8.09006 

log  a' =  1.62634    .*.  «'  =  42'.300. 

When  the  angle  is  between  3°  and  5°,  we  may  find  S'  and  T'  from 
page  24  after  finding  the  angle  approximately  from  pages  29  and  30. 
Thus  in  finding  the  angle  whose  logarithmic  tangent  is  8.77237  we 
find  from  page  29  that  the  angle  is  between  3°  23' and  3°  24',  being 
nearer  3°  23'.    Then  on  page  24  we  have 

r'=:  6.46423 
log  tan  ot  =  8.77237 
/       .*.  log  tan  a  —  7"' =  log  «'=  2.30814    /.  a'=  203'.30  =  3°  23'.3o. 

^  -CJL-v/yv  -4  \^         Angles  Greater  than  90°.^*  M^O*"^    ' 

39.  To  find  the  logarithmic  sine,  cosine,  tangent,  or  cotangent  of  an 
angle  greater  than  90°,  subtract  from  the  given  angle  the  largest  multi- 
ple of  90°  contained  therein.  If  this  multiple  is  even,  find  from  the 
tables  the  logarithmic  sine,  cosine,  tangent,  or  cotangent  of  the  remain- 
ing acute  angle.  If  the  multiple  is  odd,  the  logarithmic  cosine,  sine, 
cotangent,  or  tangent,  respectively,  of  the  remaining  acute  angle  will  be 
the  function  required  ;  thus,  sin  120°  =  sin  (90°  +  30°)  =  cos  30°. 


x  = 

I.  Quadrant. 

a 

II.  Quadrant. 

90°+a 

III.  Quadrant. 
iSo'+a 

IV.  Quadrant. 
270'+ a 

%\X\X   — 

+  sin  a 

+  cosa 

—  sin  a 

—  cosa 

cos.r  = 

+  cosa 

—  sin  a 

—  cosa 

+  sin  a 

tan  X  = 

+  tana 

—  cot  a 

+  tana 

—  cot  a 

cot  X  = 

-f  cot  a 

—  tana 

+  cot  a 

-  tana 

Or  we  could  find  the  difference  between  the  angle  and  180°  or  360°, 
and  find  from  the  tables  the  same  function  of  the  remaining  acute 
angle  ;  thus,  cos  300°  =  cos  (360°  —  60°)  =  cos  60°,  etc. 


x  = 

I.  Quadrant. 
a 

II.  Quadrant. 
180° -a 

III.  Quadrant. 
i8o'+tt 

IV.  Quadrant. 
360*— a 
or  -a 

sin  X  = 

COSJf  = 

tan  X  = 
cot  X  = 

+  sin  a 
+  cosa 
+  tana 
+  cot  a 

+  sin  a 

—  cosa 

—  tan  a 

—  cot  a 

—  sin  a 

—  cosa 
4-  tana 
+  cot  a 

—  sin  a 
+  cosa 

—  tan  a 

—  coto 

To  indicate  that  the  trigonometric  function  is  negative,  n  is  written 
after  its  logarithm. 


xii  EXPLANATION  OF   THE   TABLES. 

40.  To  find  the  angle  corresponding  to  a  given  function,  find  the 
acute  angle  a  corresponding  thereto,  and  the  required  angle  will  be  «, 
1 80°  ±  «,  or  360°  —  a,  according  to  the  quadrant  in  which  the  angle 
should  be  placed. 

41.  There  are  always  two  angles  less  than  360°  corresponding  to  any 
given  function.  Hence  there  will  be  ambiguity  in  the  result  unless 
some  condition  is  known  that  will  fix  the  angle ;  thus,  if  the  sine  is 
positive,  the  angle  may  be  in  either  of  the  first  two  quadrants,  but  if  we 
also  know  that  the  cosine  is  negative,  the  angle  must  be  in  the  second 
quadrant. 

Given  One  Function  of  an  Angle,  to  fijid  Another  without  Jindifig 

the  Angle. 

42.  Suppose  log  tan  a  =  9.79361,  and  log  cos  a  is  sought.  On  page 
57  the  tabular  difference  for  log  tan  «  is  28,  and  that  for  log  cos  « 
is  8,  the  given  logarithmic  tangent  exceeding  9.79354  by  7,  Hence 
28  :  7  =  8  :  a:  ;  .*•  jc  =  2^  X  7  =  2  =  correction  to  9.92905,  giving 
log  cos  a  =  9.92903. 

In  the  margin  are  tables  to  facilitate  the  process.  In  the  column 
headed  ^V,  the  numerator  is  the  tabular  difference  for  the  column 
headed  Z.  Cos,  and  the  denominator  that  for  the  logarithmic  tangents. 
The  numbers  in  the  marginal  column  are  the  values  of  A,  —  the  excess 
of  log  tan  a  over  the  next  smaller  logarithmic  tangent,  found  in  the 
tables,  —  such  that  ^^^A  shall  be  0.5,  1.5,  2.5,  etc. ;  and  the  numbers 
on  the  left  are  the  corrections  x  to  be  appHed  to  the  numbers  in  the 
column  headed  L.  Cos.  Thus,  if  A  is  between  1.8  and  5.2,  x  is  between 
0.5  and  1.5,  and  is  called  i.  Note  that  i  is  opposite  the  space  between 
1.8  and  5.2.  In  the  example  above,  the  excess  A  is  7,  which  lies 
between  5.2  and  8.8  ;  hence  x  is  2,  the  number  on  the  left  opposite  the 
space  between  5.2  and  8.S. 

For  example,   if  we  have   given  the  logarithms  of  the  sides  of  a 

right-angled  triangle,  \oga  =  2.98227  and  log /^  =  2.90255,  to  find  the 

hypotenuse,  we  use  the  formulas 

a  /T  /> 

tan  a  =  -,  and  e  = 


b  sin  a      cos « 

The  value  of  log  tan  a  being  found  in 

\oga  =  2.98227  (t)         the  column  marked  Z.  Tan  at  the  bot- 

logsin«  =9.88571  (4)         torn,  the  right  column  will  contain  the 

log/!' =  2.90255  (2)         logarithmic    sine    of    the  corresponding 

log  tan  «  =  0.07972  (3)         angle.     Also,  the  correction  to  9.88563 

.-.    log<r  =  3.09656  (5)         is  20  X  ^l,  which  we  find  to  be  8  from 

the  table  headed  ^.- 


j^r-'-^n 


EXPLANATION   OP^   THE   TABLES.  xiii 

I      <^=(^^^<\^'^     TABLE   in.      ^^,-.  l^''^^'^ 

43.  This  table  (pages  72-94)  contains  the  "natural"  or  actual 
numerical  values  of  the  trigonometric  sines,  cosines,  tangents,  and 
cotangents  of  angles  from  0°  to  90°,  for  each  minute,  while  Table  II. 
contains  the  logarithms  of  these  numbers. 

NoTii. : — The  secant  is  the  reciprocal  of  the  cosine,  and  the  cosecant  of  the  sine. 

The  arrangement  and  method  of  using  the  table  are  the  same  as 
those  for  Table  II.,  except  that  the  tabular  differences  are  not  printed. 
For  the  sake  of  clearness  the  first  figures  of  the  functions  are  given  only 
opposite  each  fifth  minute,  and  also  when  they  change.  Arts.  26,  27, 
29,  30,  31,  32,  and  34*  will  explain  the  table  if  the  word  "logarith- 
mic" be  changed  to  "natural,"  and  "Z.  ^///,"  etc.,  to  "iV.  Sin,"  etc. 

sin  20°  10'  =  0.34475  ;    cos  20°  10'  =  0.93869  ; 
tan  20°  10' =  0.36727  ;    cot  20°  10' =  2.7228. 
sin  68°  10'  =  0.92827  ;    cos 68°  10'  =-0.37191  ; 
tan  68°  10' =  2.4960    ;    cot  68°  10' =  0.40065. 

In  finding  sin68°24'.4  we  see  that  the  required  sine  lies  between 
0.92978  and  0.92988,  the  tabular  difference  being  10;  the  correction 
for  o'.4  is  10  X  .4  =  4  ;  hence  sin  68°  24'.4  =  0.92978  -f  4  units  in  the 
fifth  place  =  0.92982. 

In  finding  cot  68°  2o'.4  we  see  that  the  required  cotangent  lies 
between  0.39727  and  0.39694,  the  tabular  difference  being  33  ;  the  cor- 
rection for  o'.4  is  33  X  .4  =  13.2  ;  hence  cot  68°  2o'.4  =  0.39727  —  13 
units  in  the  fifth  place  =  0.39714. 

Note. — The  correction  is  added  fQ^^5  gip.g.g-JP.^l  tangent  because  they  increase  as 
the  angle  increases,  anc\  subtracted  for  the  cosine  and  cotari^nl"51fll!lB"rtiLjFii«kcrease 

in  tJT  iingin  inrrttfifrti. 


aumtffuf.-Mim 


In  finding  the  angle  whose  tangent  is  0.39870,  the  required  angle  will 
lie  between  21°  44'  and  21°  45',  the  tabular  difference  being  39896  — 
39862  =  34,  while  the  given  tangent  exceeds  that  of  21°  44'  by 
39870  —  39862  =  8.  Hence  8  —  34  or  o'.2+  is  the  correction  to  be 
added  to  21°  44',  giving  21°  44'.2  for  the  angle  required. 

In  finding  the  angle  whose  cosine  is  0.36850,  the  required  angle  will 
lie  between  68°  22' and  68°  23',  the  tabular  difference  being  36867  — 
36839  =  28,  while  the  given  cosine  is  less  than  cos  68°  22'  by  36867  — 
36850  or  17.  Hence  17-7-28  oro'.6+  is  the  correction  to  be  added 
to  68°  22',  giving  68°  2  2 '.6  for  the  angle  required. 

♦  The  examples  in  these  articles  apply  only  to  Table  II. 


XIV 


EXPLANATION   OF   THE   TABLES. 


TABLE    IV. 


44.    Circular  arcs  with   radius  unity,     (Page  95.) — To   find    the 
length  of  the  arc  of  61°  42'  35 ".2  in  a  circle  whose  radius  is  200  feet, 
we  find  that  in  the  circle  whose  radius  is  unity, 
Arc  of  61°   =  1.06465  08 
Arc  of  42'    =0.0122173 
Arc  of  35"  =  0,00016  97 
Arc  of  o".2  =  0.00000  10 
.-.    Arc    of   61°  42'  35 ".2  =  1.07703  88*   feet    in  the   circle  whose 
radius  is  i  foot,  and  if  the  radius  is  200  feet  the  length  of  the  arc  will 
be  1.07703  ZZ  X  200. 

To  find  the  angle  at  the  center  of  a  circle  with  radius  3,  the  length 
of  its  intercepted  arc  being  13.39410  00  :  the  length  of  its  intercepted  arc 
in  the  circle  whose  radius  is  unity  will  be  ^  X  13.39410  00  ■=  4.46470  00. 

4.4647000 
Next  less   =3.14159  27     corresponding  to  180°. 
Difference  =  1.32310  73 
Next  less   =  1.30899  69     corresponding  to  75°. 

.01411  04 

.01396  26     corresponding  to  48'. 

.00014  7^ 

.00014  54     corresponding  to  30". 

.00000  24     corresponding  to  o".5. 
•V  255°48'3o".5- 
*  Owing  to  the  incompleteness  of  the  table  the  last  figure  will  probably  be  erro- 


c'lf/ 


n. 


•^--vr^   -- 


•J"/  If^ 


cA^i}  t  %  1 


p/7i>m'^-// 


s^. 


EXPLANATION   OF  THE   TABLES. 


— ^j; J^-\] 


TABLE   V. 


45.    Conversion  of  common  logarithms  into  Naperian^  and  vice  versd 
(page  95) .  —  We  have 

\Q%y^N=M\o%,N,  and  \og,  N  =  ^\og,^ N . 

Table  V.   contains   the   multiples   of  M  and  —  by   numbers   from 
I  to  100. 

Find   the  common   logarithm   of   2,  its   Naperian   logarithm  being 
0.69314  718056. 

J/ X  .69  =0.2996631925 

J/ X  .0031  =    .001346312894 

J/ X  .000047  =    .000020411841 

J/ X  .00000018        =    .000000078173 

M  X  .0000000005    =    .00000  00002  1 7 

M  X  .00000000006  =    .oooQo  00000  26 

/.    Iogio2  =0.301029995651 

(True  value  =  0.30102  99957) 

Find  the  Naperian  logarithm  of  0.2,  its  common  logarithm  being 
9.30102  99957  —  10.     Hence  the  true  logarithm  is 

logio  0.2  =  —  I  H-  .30102  9995 -]  =  —  0.69897  00043. 


=      1.58878  37142 

=        .02049  30073  28 

X  .000070         =       .00016  1 1809  57 

X  .0000000043  =       .00000  00099  01 
/.  log^  0.2  =  —  1.60943  79123  86 

(True  value  =  —  1.60943  79124) 


—  X  .0080 

M 


M 

I 

'm. 


Typography  by  J,  S.  Gushing  &  Co.,  Norwood,  Mass. 


? 


C»S^  \  Vo  -r     e^i^rw  "^  (^.Je  3  '^^ 


\  y 


K 


•>. 


(^ 


(^' 


(L 


u- 


/■ 


O- 


*»^  A      i-JTS 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
This  book  is  DUE  on  the  last  date  stamped  below. 
Fine 


a' 

o 


OECJ     19521.1' 
3Har'55JP 


fiiho{,Jil^^lM 


h\ 


10ct'55KC 
SfP  17  1955  LU 

REC'D  LD 

MAY  3   195a 


REC'D  LD 

JAN  16  1959 

2Feb'59MJ 


RECD  LD 

JUL  1 6  1959 


5sl6)4120 


YC  76228 


